This thesis seeks to further investigate the quality factor which is
reported in the research literature on financial market anomalies and related
to systematic investment strategies. The research is done first by a literature
study, then by examining
whether the findings from Asness, Frazzini and Pedersen’s paper “quality minus
junk”
The first
finding on the positive relationship between price and quality shows that the
model specification based on modern asset pricing has explanatory power on
stock prices, but most of the cross-sectional variation in prices is still
unexplained. This finding is true in 40 out of the 44 countries examined for
the sample data between 2005 and 2019.
The second
finding shows that a factor-mimicking portfolio (QMJ) going long on the highest
quality firms and shorting the low-quality stocks earns a significant
risk-adjusted return with a Sharpe ratio after hedging for other factor
exposures just above 1. The risk-adjusted alpha was positive for 43 out of the
44 countries in the sample.
This thesis
contributes the empirical asset pricing field by confirming the results from
Asness et al. using a broad sample of 44 countries obtained from a different data
provider and with all the factors built from scratch. In their paper they
conclude the abnormal risk-adjusted returns of quality stock are due to
mispricing and they are unable to find a risk-based explanation. This study
supports those conclusions and I find that quality deliver consistent returns
during times of distress as well as in times of boom. It is difficult to find a
risk-based explanation of the abnormal returns or a behavior-based story for
why investors underweight high-quality stocks. Rather, the intelligent investor
should add the QMJ factor to his or her toolbox of factors which can be used to
create a portfolio aligned with the investor’s goals and preferences.
An
investment operation is one which, upon thorough analysis, promises safety of
principal and a satisfactory return. Operations not meeting these requirements
are speculative.
–Benjamin Graham (1934)
Contents
2.1 A Brief History of Asset
Pricing
2.2.2 Expected Return-Beta Model
2.2.3 Selecting factors – market anomalies
3.1 Quantopian Research Platform
4.2 Correlations Between Factor
Components
4.5 The Price of Quality Stocks
4.5.2 Price of Quality in the Cross-Section
4.6 The Return of Quality Stocks
4.6.1 The Return of Univariate Portfolio Sorts on Quality
4.6.2 The Return on the QMJ Factor
4.7 Time-varying Return of
Quality
Appendix 3 – Python notebook with
code used to perform analysis.
List of Figures
Figure 2‑1: Relationship of Intrinsic value factors to
market price.
Figure 4‑1: Quality plotted against its sub-factors
profitability, growth and safety.
Figure 4‑2: Pair-plots of the profitability factor and its
sub-components.
Figure 4‑3: Pair-plots of the growth factor and its
sub-components.
Figure 4‑4: Pair-plots of the safety factor and its
sub-components.
Figure 4‑5: Portfolio mean return versus cross-sectional
regressions [23].
Figure 4‑6: This chart shows the high-minus low of quality
sorted portfolios
Figure 4‑7: Change in mean portfolio quality scores from time
of formation
Figure 4‑8: Typical scatterplot of time averaged sample data
and fitted regression line
Figure 4‑13: QMJ 4-factor alpha information ratios.
Figure 4‑14: The chart shows the Global average cumulative
excess returns from the factor portfolios
Figure 4‑15: The time-varying price of quality.
Figure 4‑16: This chart shows the cumulative 4-factor alpha
factor abnormal returns
Figure 4‑18: Cumulative excess returns of a QMJ portfolio
formed every month.
Figure 4‑19: Workflow for quantitative investment, adopted
from [31]
List of Tables
Table 2‑1:
Variable definitions
Table 3‑1: Summary statistics of data sample
Table 4‑1: Pearson product-moment correlations between
quality and sub-factors
Table 4‑2: The price of quality - Cross sectional regressions
Table 4‑3: Return on Quality - excerpt from Table A-2 showing
two countries of interest
Table 4‑4: Regressions of QMJ returns on risk factors.
The goal of
this thesis is to examine factor models used in the asset pricing of stocks.
More specifically, I wish to determine whether the findings relating to the
quality factor proposed by Asness, Frazzini and Pedersen
This puzzle
intrigued me to dig into the field of empirical asset pricing and learn the
skills to put factor models to the test. In order to prove or disprove the
findings from Asness et al. I have formulated these two hypotheses that need to
be tested:
1) There is a positive, but weak,
correlation between asset price and a firm’s quality.
2) A significant risk-adjusted return
can be earned by investing in (going long) high-quality stocks and shorting the
low-quality stocks.
The problem of efficient allocation of limited resources is a fundamental issue
to understand in economics. As student we are introduced to utility theory as
model to understand how people use their resources efficiently by taking into
account preferences, e.g. to risk or other factors. Utility theory is used to expand
into the modern portfolio theory. Markowitz’s minimum variance problem and the
capital asset pricing model (CAPM) are presented as frameworks to understand pricing
of individual assets under market equilibrium. This framework works well as an
academic model, but given its many unrealistic assumptions the traditional derivations
do not hold up against empirical data
Since the CAPM was developed in the 1960s alternatives and improvements have
been proposed by several researchers. Ross developed the alternative arbitrage
pricing theory (APT)
The typical
textbook economic theory teaches us that stock prices in the market fully
reflect all available information, sometimes referred to as the efficient
market hypothesis
“The market
is not a weighing machine, on which the value of each issue is recorded
by an exact and impersonal mechanism, in accordance with its specific
qualities. Rather should we say that the market is a voting machine,
whereon countless individuals register choices which are the product partly of
reason and partly of emotion.”
Graham & Dodd were also some of the first authors to
break down factors that affect the market prices and in doing so they made a
sharp distinction between what they called speculative factors and analytical
(or investment) factors. The purpose of including Figure 2‑1 is to show that
there is long timeline from those early “experience-based” observations of
market factors to the more quantitative studies of recent years and attempts to
explain them for instance by behavioural economics and factor models. Some of these
explanations of how humans don’t act like the rational Homo Economicus, like
prospect theory, have have been popularized through the book “Thinking, fast
and slow”
Figure 2‑1: Relationship of Intrinsic value factors to market price.
From the 1934 textbook Security Analysis
In this
thesis I reference the textbooks by two leading scholars, John Cochrane of
University of Chicago
The fundamental
concept in all asset pricing, from stocks to bonds to options, is this: price
equals the expected discounted payoff. The investor must choose how much to
consume now and how much to save for tomorrow. The marginal utility loss of consuming
less today and buying some asset should equal the marginal utility gain of consuming
more of the asset’s payoff in the future. If the price and the future payoff
does not satisfy this condition the investor will either buy more of or sell
the asset. The investor’s first order condition for optimizing that choice leads
to the consumption-based asset pricing model:
In the case
of a stock the expected payoff x at a given time is the expected price plus the
expected dividend payment at this given time. We treat the payoff as a random
variable which can take many possible outcomes. The utility function u may take any form we’d want, e.g.
As a side note, most
asset pricing models, like CAPM, ICAPM or APT, can be derived as special cases
from the pricing equation (1) by imposing different constraints and form to the
stochastic discount factor. For example, in the CAPM the discount factor
Consider
that we have a certain risk-free rate; then the discount factor becomes
These derivations
lead to the fundamental insight that the asset’s price and its expected excess
return (risk premium) depend whether the payoff/return covary positively or
negatively with the investor’s stochastic discount factor. As consumption c increase, the marginal utility m declines (diminishing return). If the asset payoffs x
also declines together with m it implies a higher asset price. If payoffs
covary negative with m investors will be willing to pay a lower price.
The insight can be explained from risk aversion; if
the investor holds an asset that has a positive covariance with consumption,
i.e. pays off well when you feel rich and pays less when you feel poor, it will
make the consumption stream more volatile. A negative covariance between
returns on the other hand will reduce consumption volatility and the investor
can keep a steady consumption even in bad times. Essentially, we value assets
that pay us when we are most “hungry” for money. The variance of the asset
payoffs themselves are irrelevant and does not generate a risk premium; the
investor cares only about volatility in his own consumption.
Traditional asset
pricing models, like CAPM, ICAPM and APT, often measure the investor’s “hunger”
by evaluating the behavior of large asset portfolios. This evaluation is done
by manipulating the pricing equation above to allow representing expected
return by betas which are suitable for linear regression (note removal of
excess return and that time subscript is removed):
Worth noting here
is that the l is not asset specific and often interpreted as the
price of risk, whilst the b is
asset specific and the quantity of risk in each asset. Gamma is the inverse of
E(m). For practical purposes we often wish to use
factors that are not direct measurements of consumption growth. This goal can
be achieved by introducing the concept of “factor-mimicking portfolios”, in
which we select a portfolio of assets whose payoff or return correlate closely with
the discount factor m. The payoff space X is the set of all payoffs that
investors can invest in and where investors can form any portfolio of traded
assets or linear combinations of payoff vectors. Thus, a portfolio can be
represented as vector of payoffs
This discount factor is called the
mimicking portfolio for m and is holds the same pricing implications as m, i.e.
we can substitute all the m’s in equation 5 with x*. Using the same arguments,
we can create any factor f in which the factor-mimicking excess returns is the
orthogonal projection of vector f onto the excess return space Re.
We can use the model form of equation 5 with the betas being regression
coefficients of the returns on the factor-mimicking portfolio (not the factor
itself). When using return as a factor the model becomes very elegant, since
the factor risk premium is also the expected excess return.
By expand the model and saying b and l is a
linear combination of a sum of bk and lk we end up with the expected return-beta model form we will use in this
thesis:
In the cross-sectional regression above
the bk’s are the exposure of asset i to
risk factor k and lk is the expected return for each unit of this exposure. We can also run
a time-series regression for each asset i where beta are the coefficients we
get when running a regression of return on factors. The factors i are (or
should be) proxies for marginal utility growth:
When
generating a factor-mimicking portfolio, the analyst will look for factors that
have predictive power on future returns. As described above we can trivially
fit an unlimited number of factors to suit the return space data. So, to quote
Cochrane the challenge is that:
“Most
empirical asset pricing research posits an ad hoc pond of factors, fishes
around a bit in that pond, and reports statistical measures that show
‘‘success,’’ in that the model is not statistically rejected in pricing a set
of portfolios”
So how do
we combat this? According to Cochrane the best advice is to understand
fundamental macroeconomic sources of risk, use economic theory to carefully
specify the factors applied and use cross-sample and out-of-sample checking of
your model’s stability. The purpose is not necessarily to have a perfect data
fit, but to describe how the investor’s “hunger” varies along axes of interest
in the cross-section and/or in time.
Factors that
do not fit into the effective market hypothesis or CAPM framework have
historically been called market anomalies, styles or risk factors. William
Sharpe
Hou et al.
|
Factor |
Description |
|
Momentum |
The phenomenon that securities which have performed well relative
to peers (winners) on average continue to outperform, and securities that
have performed relatively poorly (losers) tend to continue to underperform |
|
Value |
Value is the phenomenon that securities which appear “cheap”
on average outperform securities which appear to be “expensive” |
|
Investment |
A negative relation between capital investments for a firm
and its future returns. |
|
Profitability |
An observation that more profitable firms have higher
expected returns than less profitable profitable firms. |
It is
interesting to note that even with all the developments in data availability
and computing power it seems like many of the experience-based guidelines
Benjamin Graham gives in his book “The intelligent investor”
In order to
avoid the pitfalls of selecting factor models that result from data mining or
other misuse of statistics and data several authors have in recent years
provided guidelines and heuristics. Both Hsu et al.
· Establish an ex-ante economic
foundation
· Factors should be robust across
definitions and geographies (cross-validation)
· Do not ignore trading costs and fees
· Ensure good data quality and
document data transformations
As ever, it
pays to listen to the advice of the old masters, here from Albert Einstein
(1933):
“It can scarcely be denied that the supreme
goal of all theory is to make the irreducible basic elements as simple and as
few as possible without having to surrender the adequate representation of a
single datum of experience.”
In this
thesis I will focus on investigating the factor, or investing style, called
quality. It refers to a hypothesis (and finding) that investing in highly
profitable, operationally efficient, safe and stable companies tend to
outperform the market over time. There is no good common definition of quality
across the literature, but Asness et al.
Then, what
would those characteristics be? NBIM
· Profitability
The economic reasoning is that, all else equal, highly profitable companies
should command a higher stock price. Profitability refers to the ability to
generate earnings compared to expenses and can be measured by many accounting
ratios. Hou et. Al
· Safety
The basis of safety characteristic is that, all else qual, investors should a
higher price for companies with a lower required return (when looking at the
companies’ discounted cash flow). Risk of default by for instance excessive
leverage would by economic theory increase the financing cost of a firm and
thus the required return. Typical factors that describe the safety of a company
are often related to a strong balance sheet, like low debt-to-assets, high
current ratios, or volatility of profitability factors.
· Growth
Growing profits are considered a characteristic that investors should pay a
premium for, all else equal. This growth can typically indicate that the
company has a sustaining competitive advantage over the competition. It can be
measured as X-year growth in profitability (measured as above) or considering
volatility over time.
Additionally,
it is important to bring up research that failed to find a statistically
significant “quality factor”. Beck et al
Asness et
al. derive a mathematical model for quality using the firm value (price)
described as the present value of all future dividends as a starting point (as described
in section 2.2):
The
important point to note here is their choice of stochastic discount factor
(also called pricing kernel), in which
Value
profitability growth
safety (negative risk)
Using this
model founded on “first principles”, we see that the firm’s value can be
explained by factors relating to its ability to generate profit, growth and
avoid negative risk, which was hypothesized in the previous chapter. Other
authors, e.g. Frama and French (2014)
The next
step in developing the factor model is to find representative proxies for the
variables in the equation above. We are not looking to price assets correct in
absolute values, but to compare prices relative to each other. Asness et al.
use the result from robust studies to select available fundamental data points
for companies and to construct proxies for profitability, growth and safety. For
instance, the profitability measures used have been mostly selected from a
highly cited study by Novy-Marx
The following
table provides a summary of the variables. The growth variables with a
delta-prefix indicates the 5-year change of the variable. To see how I have
constructed these measures in the analysis I refer to the Python notebook’s
section 2.2 in Appendix 3.
Table 2‑1: Variable definitions
|
Variable |
Description |
Variable |
Description |
|
GPOA |
Gross profit over assets |
BAB |
Market beta |
|
ROE |
Return on equity |
LEV |
Leverage (debt over assets) |
|
ROA |
Return on assets |
O |
Ohlson’s O-score |
|
CFOA |
Cash flow over assets |
Z |
Altman’s Z-score |
|
GMAR |
Gross margin |
EVOL |
Earnings volatility |
|
ACC |
Fraction of cash earnings |
|
|
For all the
data analysis in this thesis I have used Quantopian.com, which provide a
cloud-based data science platform for performing quantitative financial
analysis using the Python programming language. The platform provides an IDE
(interactive development environment) to perform research on equity data and an
engine to perform backtesting of trading algorithms.
The data
available through Quantopian contain quality checked equity data from 44
countries from 2004-2019. For example, to avoid survivorship bias they contain
data as stocks are listed and delisted and stored point-in-time so that the
backtesting simulation engine avoids any lookahead bias. As an example, in
asset pricing research it is fairly customary to construct factors based on
accounting data with a six months’ time lag to avoid lookahead bias. With
point-in-time data this is not necessary because data in the backtesting engine
will only become available in the simulation at the historical filing date of
each company’s financial reporting.
The data
sources used in this thesis are from FactSet and contain fundamentals data,
equity pricing and metadata and RBICS (business industry classification). To
expand the analysis the available data also range from analyst estimates to insider
trade transactions and news sentiment.
Although
the data sources are of high quality, they still must be processed in order to
obtain sample data that can be analyzed and that are relevant. The first data
screening is to remove any non-tradable assets and to only include primary
shares. The primary share is defined as the first share/ticker that a company
has at IPO and is still actively trading. If this share is no longer trading,
the share with the highest volume is denoted as the primary share. After this
initial filtering, we typically see that the datasets contain up to 10-30% of
missing data for some of the accounting/fundamental data that we use to build
the quality factor. This is typically the smaller stocks that we anyway almost
disregard when value-weighting the components later in the analysis work. Some
data is critical, so any stock with missing market to book ratio has been
completely removed.
The next
step of processing is to build all the quality factors given in section 2.3.2. First, we perform a winsorization to
limit the effect of extreme values and possible spurious outliers (especially relevant
for accounting data and ratios). 95% winsorization is done by setting all
values outside the lower and upper bound (2.5 and 97.5 percentile) equal to the
boundary value.
Next, we know
that accounting data is different when comparing across industries. For
example, the profit margin is typically much higher for companies in the
financial sector than in the utilities sector (although this tells us little
about the returns from investing in either industry). To normalize accounting
data across industries we demean each factor my sector. This means that the
most profitable investment bank is ranked equal as the most profitable utility
company when constructing the profitability factor. There are several choices
on methodology for industry demeaning and each one has its positive and
negative sides. Some researchers eliminate all financial services firms from
the analysis, they cluster regressions by industry or they enforce sector
neutrality by weighting methods. For this work I have chosen to demean by grouping
all stocks within each sector together and then normalizing each of the factor
components within sectors. The dataset groups each asset in one of 13
industries, but unfortunately industry classification is missing for a lot of
the smaller markets (again, this is most widespread in small stocks). Comparing
results before and after industry demeaning showed a marked improvement when
inferring statistical significance of the regressions.
When
normalizing (z-scoring) the fundamental data we assign the stocks with missing
data to have a score of zero, such that when we aggregate the normalized
factors the missing data for each stock is effectively ignored. The method may
not be perfect, but for empirical research it serves the purpose of not having
to delete every stock which is missing some data which would reduce our sample
data. Finally, the stocks are ranked and then normalized as suggest by Asness
et al. When the procedure in this section has been performed, we are left with
zero missing data for the analysis work.
The 44
countries analyzed are listed below, sorted by size along with some summary
statistics. All the countries have been analyzed from 2005 until June 2019 and
consist of a total of 49 003 companies.
The market
capitalization time series is converted to US dollars using London Market spot
exchange rates at close of each day and the mean value here is across the
entire time series. The number of stocks per month is the number included in
each monthly calculation and the total stocks is the total number of unique
assets over the entire time period. The global market weight is a naive
calculation of each countries relative size as the weighted average of the mean
market cap multiplied by number of stocks per month. For comparison I have also
made a column for each countries weight based on the number of stocks in each
universe divided by total number of stocks.
Table 3‑1: Summary statistics of data sample
|
Country |
Mean Market Cap |
Stocks per month |
Total Stocks |
Global Market Weighted Size |
Global Equal Weight Size |
|
United States |
4.15E+09 |
4502 |
6866 |
34.9 % |
14.0 % |
|
Great Britain |
1.78E+09 |
1723 |
3820 |
8.3 % |
7.8 % |
|
Japan |
1.17E+09 |
3723 |
5092 |
7.3 % |
10.4 % |
|
Hong Kong |
2.32E+09 |
1454 |
2493 |
7.1 % |
5.1 % |
|
China |
1.30E+09 |
2143 |
3575 |
5.7 % |
7.3 % |
|
Canada |
6.59E+08 |
2554 |
4887 |
3.9 % |
10.0 % |
|
Germany |
1.80E+09 |
846 |
1503 |
3.3 % |
3.1 % |
|
Australia |
6.99E+08 |
1680 |
2866 |
2.5 % |
5.8 % |
|
South Korea |
6.54E+08 |
1571 |
2620 |
2.1 % |
5.3 % |
|
Switzerland |
4.40E+09 |
242 |
374 |
2.0 % |
0.8 % |
|
Russia |
2.78E+09 |
236 |
543 |
1.9 % |
1.1 % |
|
Spain |
4.67E+09 |
158 |
319 |
1.8 % |
0.7 % |
|
Sweden |
1.17E+09 |
485 |
1132 |
1.6 % |
2.3 % |
|
Taiwan |
4.88E+08 |
1647 |
2379 |
1.4 % |
4.9 % |
|
Brazil |
2.92E+09 |
174 |
283 |
1.0 % |
0.6 % |
|
Netherlands |
4.10E+09 |
107 |
201 |
1.0 % |
0.4 % |
|
South Africa |
1.28E+09 |
312 |
602 |
0.9 % |
1.2 % |
|
Singapore |
6.94E+08 |
678 |
1018 |
0.9 % |
2.1 % |
|
Mexico |
3.17E+09 |
116 |
196 |
0.8 % |
0.4 % |
|
Norway |
1.07E+09 |
241 |
500 |
0.7 % |
1.0 % |
|
Indonesia |
7.42E+08 |
428 |
660 |
0.6 % |
1.3 % |
|
Malaysia |
3.77E+08 |
943 |
1294 |
0.6 % |
2.6 % |
|
Denmark |
1.61E+09 |
169 |
280 |
0.6 % |
0.6 % |
|
Thailand |
5.23E+08 |
564 |
834 |
0.5 % |
1.7 % |
|
Finland |
1.67E+09 |
128 |
203 |
0.4 % |
0.4 % |
|
Turkey |
6.68E+08 |
321 |
472 |
0.4 % |
1.0 % |
|
Poland |
3.46E+08 |
442 |
760 |
0.3 % |
1.6 % |
|
Colombia |
2.64E+09 |
47 |
82 |
0.3 % |
0.2 % |
|
Austria |
1.69E+09 |
76 |
127 |
0.3 % |
0.3 % |
|
Philippines |
6.98E+08 |
230 |
303 |
0.3 % |
0.6 % |
|
Argentina |
1.41E+09 |
78 |
116 |
0.2 % |
0.2 % |
|
Ireland |
2.48E+09 |
33 |
64 |
0.2 % |
0.1 % |
|
Greece |
3.78E+08 |
243 |
363 |
0.2 % |
0.7 % |
|
Portugal |
1.44E+09 |
52 |
89 |
0.2 % |
0.2 % |
|
Peru |
6.16E+08 |
120 |
191 |
0.1 % |
0.4 % |
|
Czech
Republic |
1.97E+09 |
19 |
55 |
0.1 % |
0.1 % |
|
New Zealand |
4.26E+08 |
126 |
223 |
0.1 % |
0.5 % |
|
Pakistan |
2.14E+08 |
246 |
342 |
0.1 % |
0.7 % |
|
Hungary |
7.23E+08 |
35 |
64 |
0.1 % |
0.1 % |
In order to
answer my three main research questions and to replicate the “Quality minus
junk” study I will perform the following analyses to test each hypothesis:
1) There is a positive correlation
between price and quality
a. Persistence of quality
b. Regression of price on quality
2) There is a positive risk-adjusted
return from investing in (going long) high-quality stocks and shorting the
low-quality stocks
a. Regression of excess return on
quality sorted portfolios
b. Regressions of the QMJ
factor-mimicking portfolio returns on risk factors.
The
following plots show the relationship between the quality factor and its
sub-factors profitability, growth and safety and the relationship between each
sub-factor and their individual components. The pairwise correlations for the global
sample mean weighted by market size is found in Table 4‑1. The main finding you can see
graphically from Figure 4‑1 is the strong pairwise correlation
coefficients between the quality components. The correlation between profitability
and growth of 0.67 across the whole global value weighted sample indicate that
profitability is persistent and this is in line with findings from Novy-Marx
From Figure 4‑2 to Figure 4‑4 we see that there looks to be a
positive relationship between all the components (fundamental data) that make
up a factor, which tells us that Asness et al. found a robust set of proxies
for their factors. By robust I mean that if some data constituting e.g. the
safety factor is missing or has measurement error it will have less effect on
the aggregated main factors. It should also be noted that the factor
sub-components are not shown in their normalized form and that the underlying sample
distributions of fundamental data are very much non-normal.
Table 4‑1: Pearson product-moment correlations
between quality and sub-factors
|
|
Quality |
Profitability |
Growth |
Safety |
|
Quality |
1.00 |
0 |
||
|
Profitability |
0.82 |
1.00 |
||
|
Growth |
0.73 |
0.62 |
1.00 |
|
|
Safety |
0.48 |
0.12 |
0.07 |
1.00 |
Figure 4‑1: Quality plotted against its sub-factors profitability, growth and safety.
The diagonal shows the sample distribution of
each factor along with a fitted regression line to indicate direction of
relationship. Pearson product-moment correlations are denoted by R-value. Data
is US sample from ’05-‘19.
Figure 4‑2: Pair-plots of the profitability factor and its sub-components.
The diagonal plots are the sample
distributions and the off-diagonal scatter plots also show a fitted
regression-line to indicate direction of relationship. Pearson product-moment
correlations are denoted by R-value. The sample is US stocks ’05-’19
Figure 4‑3: Pair-plots of the growth factor and its sub-components.
The diagonal plots are the sample
distributions and the off-diagonal scatter plots also show a fitted
regression-line to indicate direction of relationship. Pearson product-moment
correlations are denoted by R-value. The sample is US stocks ’05-’19
Figure 4‑4: Pair-plots of the safety factor and its sub-components.
The diagonal plots are the sample
distributions and the off-diagonal scatter plots also show a fitted
regression-line to indicate direction of relationship. The sample is US stocks
’05-’19
In the
following sections we run regression analyses on price and on returns. Our main
tool is ordinary least-squares regression, but we use the procedures of
Fama-Macbeth to obtain corrected standard errors and correct for
autocorrelation using the method of Newey-West. Although these are the traditional
tools used in asset pricing research (and by Asness et al.) the more modern
approach, summarized nicely by Peterson
Despite
this, because I struggled to implement clustered errors in the Python
Statsmodels regression library (and to stay true to Asness et al.’s
methodology) I chose to use the Fama-Macbeth procedure described here.
The
procedure is slightly different depending on what our factors are. For
observable characteristics the first step below is often omitted or taken as a
separate analysis. In our case we do regressions on quality, which is
calculated separately for each asset and each time step. So, there is no need
to estimate asset-specific betas using equation 14 below.
The
Fama-Macbeth two-step regression procedure is often used in analysis of factors
that explain asset returns. It is a practical “two-pass” way of testing how the
factors describe portfolio or asset returns by finding the return premium from
exposure to the factors. First, each portfolio’s or asset’s return is regressed
against the factor time series
In a
“traditional” two-step regression, we would then use equation (15) to estimate
a single cross-sectional regression with the sample averages, but Fama-MacBeth
suggested that instead we run a cross-sectional regression at each time period. The main advantage of Fama-MacBeth is to then
average these coefficients, once for each factor, to give the premium expected
for a unit exposure to each factor (19) and alpha (17) over time. This method
splits the sample into T smaller samples and we can deduce the variation across
samples (time), assuming no autocorrelation. Our estimators simply become the
average across time (sample mean) and the sampling errors are generated from
the standard deviation of the sample means:
If the
prices are independent and identically distributed (iid) normally over time,
then the t-statistic can be used to test the null hypothesis that the
regression coefficients are zero. See Cochrane
Before digging into the analysis of quality-sorted portfolios, I will
briefly explain the method used. Portfolio analysis is traditionally a very
commonly used method in empirical asset pricing to examine the cross-sectional
relationship between some variable(s). It is essentially a non-parametric
cross-sectional regression using non-overlapping histogram weights, as
illustrated in Figure 4‑5. The big motivation for creating portfolios is
to remove the “noise” (idiosyncratic volatility) of each individual asset by
bundling them into portfolios of assets that have relatively similar exposure
to a factor. The univariate portfolio analysis procedure has, as detailed
nicely by Bali et al.
1)
Calculate the factor
breakpoints that will be used to divide the sample into portfolios.
2)
Use these breakpoints to
form the portfolios.
3)
Calculate the average
value of the outcome variable Y within each portfolio for each period t and
present the time series average with corrected standard errors.
4)
Examine the variation in
these average values of Y across the different portfolios.
a.
Examine if the time-series
mean of the portfolios, especially the difference portfolio (H-L), is
statistically different from a null hypothesis mean value (often zero). A
non-zero mean is evidence that a cross-sectional relation exists between the
sort variable and outcome variable.
There are also methods for creating bivariate (double-sorted)
portfolios, which is what Asness et al. does when creating the QMJ factor by
sorting on size (market capitalization) and then sorting on the quality factor.
This bivariate sort is similar to a regression on quality controlled for size.
According to Cochrane
Figure 4‑5: Portfolio mean return versus cross-sectional regressions
In order to
determine the price of quality stocks I first perform a univariate portfolio sort
on quality to split each stock universe into ten equal-sized quality
portfolios. This is the first step of testing whether high quality firms
command higher prices. If quality is persistent it means the market can predict
future quality and take this into account when determining the prices today.
The results
from Table A-1 in Appendix 1 show us that the quality score is consistent over
time for the entire sample. This is also illustrated visually in the figures
below. Figure 4‑6 shows the difference portfolio
(high minus low) for each country in the sample at the time of portfolio
formation and three and ten years after formation. Each month we form the
quality sorted portfolios and record the quality scores of the same portfolio
three and ten years later. We examine if the time series mean of the difference
portfolio after three and ten years is statistically distinguishable from zero
as an indication/evidence that a cross-sectional relationship is persistent.
This is done by regressing the time series means on a constant and implementing
the Newey-West adjusted standard errors to correct for heteroskedasticity and autocorrelation.
We also
want to know if there is a monotonic pattern in the quality sorted portfolios. Figure 4‑7 shows the average portfolio quality
means for the entire sample. At portfolio formation the monotonicity is by
construction, but we also see that the monotonic pattern is persistent even
after ten years. There is a regression towards the mean, but we clearly see
that on average a significant number of the high-quality companies at portfolio
formation are still winners even ten years later and vice versa.
The combined
results allow us to conclude that quality is persistent is every country of the
sample and that it is possible to select companies which will exhibit high
quality in the future by looking at their recent past. In theory that should
mean that the market has the necessary information to correctly reflect future
quality in today’s prices.
Figure 4‑6: This chart shows the high-minus low of quality sorted portfolios
at time of portfolio formation and the
corresponding mean portfolio quality scores 3 and 10 years after portfolio
formation. All estimates are statistically significant; the average
t-statistics for the 3-year lagged means is 32.61 and for the 10-year lagged means
it is 15.72.
Figure 4‑7: Change in mean portfolio quality scores from time of formation
and after 3 years and 10 years. Portfolio
scores are the equal weight mean of the entire global sample.
We now run
a cross-sectional regression of price on quality. Although it varies from the
expected return-beta model its equivalence can be seen by examining equation 3 to
5.
We perform
the natural logarithmic transformation of highly skewed ME/BE data into “close
to” normally distributed data, but to avoid numerical issues I applied
Figure 4‑8: Typical scatterplot of time averaged sample data and fitted regression line
(US ’05-’19). Distribution of Samples shown on
top and right along with Pearson’s correlation number (0.27) between the two
variables
From Table 4‑2 we see that our hypothesis stating that
there is a positive correlation between price and quality cannot be rejected
for the big majority of countries. Our regressions show significant coefficient,
all our sub-components of quality have the same sign and the results are in
general in alignment with Asness et al. Although I have not included all the
controlling factors that they have I find the explanatory power of the regressions
to be in the same range as their study, with about 8% of variation explained in
the US sample and 8% in the value weighted global sample. The magnitudes of the
coefficients are in the same order of magnitude and the aggregation of sub-components
into the quality score retains both magnitude and statistical significance. Given
that our starting point was a theoretically plausible formulation of a firm’s fundamental
value it is perhaps surprising that a regression model made of the components
profitability, growth and safety explain very little of the cross-sectional
variation in prices. But nonetheless, we can conclude that higher quality
companies do command higher prices than “junk” companies.
Looking at the
magnitudes, Figure 4‑9 zooms in on the quality slop
estimate for the sample. First, we see that the magnitude varies, but the
majority of countries exhibit a statistically significant positive relation
with a value weighted sample mean of 0.11. This is interpreted such that if the
quality score of a company moves one standard deviation higher, ceteris paribus,
the market to book value (ME/BE) ratio only increases 11% from the total sample.
Or put differently; companies having a quality score in the 99th
percentile (2σ) can enjoy only a 22% higher stock price than the average company (all else
equal, particularly book equity). The slope coefficient estimated for quality is
much lower than the 0.22 and 0.24 coefficient that Asness et al. finds, but
their sample is much longer and could indicate that relationship between price
and quality is lower in more recent years. For the Norwegian reader it is worth
noting that the slope coefficient for the Oslo Stock Exchange is on the lower
side at 0.05 (with a t-statistic of 12.57), i.e. the price of quality in Norway
is low compared to the global sample.
Our
analysis on the relationship between price and quality leaves us to conclude
that our model specification for quality does explain prices, i.e. our first
hypothesis that high-quality firms exhibit higher prices cannot be rejected for
40 of our 44 countries. The findings here confirm and support the results from
Asness et al.’s study on the quality factor.
Figure 4‑9: This chart’s left-hand axis shows the slope coefficient estimates for a regression of price on quality for each country.
The crosses indicate the t-statistic for each
regression and the t-values are given on the right-hand axis scaled by log2,
which means that all the estimates with crosses below the main horizontal axis
have a t-statistic less than 2.0 (ireland and phillipines).
Table 4‑2: The price of quality - Cross sectional regressions
This table presents the results from monthly Fama-Macbeth regressions. The
dependent variable is the natural logarithms of a firm’s market-to-book ratio
plus one at time t. The explanatory variables are the quality scores at time t.
AdjR2 is the time series average of the adjusted R-squared of the
cross-sectional regressions of price on quality (model 1). Standard errors are
adjusted for heteroskedasticity and autocorrelation using a lag length of 5
periods. T-statistics are shown below the coefficient estimates in paranthesis.
The top row shows the global sample mean by market size weights. Countries as
sorted by market size from big to small.
|
|
(1) |
(2) |
(3) |
(4) |
(5) |
Adj R2 |
|
|
||
|
Quality |
Profitability |
Growth |
Safety |
Profitability |
Growth |
Safety |
||||
|
Sample mean |
0.11 |
0.08 |
0.06 |
0.11 |
0.06 |
0.02 |
0.10 |
0.06 |
171 |
2363 |
|
United States |
0.19 |
0.15 |
0.15 |
0.12 |
0.10 |
0.07 |
0.10 |
0.079 |
119 |
4133 |
|
(54.74) |
(28.84) |
(22.13) |
(23.84) |
(20.64) |
(7.28) |
(14.71) |
||||
|
Great Britain |
0.13 |
0.06 |
0.00 |
0.20 |
0.10 |
-0.04 |
0.20 |
0.028 |
172 |
1652 |
|
(26.29) |
(7.25) |
(-0.73) |
(46.75) |
(6.73) |
(-3.14) |
(40.65) |
||||
|
Japan |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.062 |
163 |
3723 |
|
(11.00) |
(13.37) |
(0.00) |
(0.00) |
(17.56) |
(-8.50) |
(0.00) |
||||
|
Hong Kong |
0.04 |
0.01 |
0.02 |
0.07 |
-0.01 |
0.01 |
0.07 |
0.022 |
168 |
1440 |
|
(29.47) |
(3.58) |
(19.38) |
(24.14) |
(-2.12) |
(2.22) |
(23.50) |
||||
|
China |
0.06 |
0.03 |
0.04 |
0.07 |
-0.03 |
0.05 |
0.08 |
0.034 |
165 |
2126 |
|
(15.05) |
(7.82) |
(11.44) |
(10.25) |
(-5.58) |
(10.32) |
(10.23) |
||||
|
Canada |
-0.01 |
-0.11 |
-0.08 |
0.18 |
-0.06 |
0.01 |
0.18 |
0.000 |
172 |
2334 |
|
(-0.37) |
(-7.23) |
(-5.81) |
(11.70) |
(-3.51) |
(0.75) |
(16.06) |
||||
|
Germany |
0.11 |
0.06 |
0.02 |
0.16 |
0.04 |
0.00 |
0.16 |
0.028 |
173 |
817 |
|
(21.38) |
(10.72) |
(3.83) |
(25.81) |
(5.28) |
(-0.23) |
(24.69) |
||||
|
Australia |
-0.01 |
-0.10 |
-0.07 |
0.15 |
-0.07 |
0.02 |
0.13 |
0.001 |
172 |
1620 |
|
(-2.13) |
(-12.78) |
(-10.96) |
(18.67) |
(-8.86) |
(2.52) |
(20.44) |
||||
|
South Korea |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.015 |
167 |
1571 |
|
(15.59) |
(0.40) |
(9.53) |
(13.79) |
(-5.74) |
(8.26) |
(11.70) |
||||
|
Switzerland |
0.14 |
0.13 |
0.07 |
0.11 |
0.15 |
-0.05 |
0.10 |
0.077 |
171 |
239 |
|
(24.00) |
(17.69) |
(10.55) |
(20.13) |
(13.43) |
(-3.87) |
(17.12) |
||||
|
Russia |
0.04 |
-0.01 |
0.05 |
0.06 |
-0.08 |
0.08 |
0.06 |
0.010 |
168 |
234 |
|
(7.30) |
(-2.09) |
(8.28) |
(8.79) |
(-8.51) |
(9.68) |
(6.34) |
||||
|
Spain |
0.18 |
0.17 |
0.09 |
0.14 |
0.17 |
-0.04 |
0.10 |
0.069 |
174 |
154 |
|
(13.05) |
(11.40) |
(6.61) |
(16.22) |
(9.57) |
(-2.61) |
(13.06) |
||||
|
Sweden |
0.02 |
-0.03 |
-0.03 |
0.09 |
-0.01 |
-0.01 |
0.08 |
0.003 |
171 |
482 |
|
(7.02) |
(-7.74) |
(-7.60) |
(19.24) |
(-1.70) |
(-1.70) |
(19.35) |
||||
|
Malaysia |
0.08 |
0.07 |
0.04 |
0.07 |
0.07 |
-0.02 |
0.05 |
0.115 |
164 |
943 |
|
(37.94) |
(36.00) |
(23.09) |
(38.65) |
(29.25) |
(-15.81) |
(28.57) |
||||
|
Taiwan |
0.02 |
0.01 |
0.00 |
0.02 |
0.01 |
0.00 |
0.01 |
0.117 |
163 |
1647 |
|
(34.43) |
(23.15) |
(14.21) |
(26.99) |
(18.84) |
(-8.31) |
(16.05) |
||||
|
Brazil |
0.17 |
0.16 |
0.06 |
0.15 |
0.19 |
-0.07 |
0.12 |
0.109 |
168 |
170 |
|
(20.66) |
(12.88) |
(7.53) |
(15.30) |
(11.55) |
(-5.64) |
(10.41) |
||||
|
Netherlands |
0.08 |
0.03 |
-0.02 |
0.16 |
0.08 |
-0.07 |
0.16 |
0.016 |
174 |
105 |
|
(5.43) |
(0.00) |
(-1.04) |
(8.28) |
(4.36) |
(-2.31) |
(7.77) |
||||
|
South Africa |
0.04 |
0.04 |
0.02 |
0.03 |
0.06 |
-0.03 |
0.02 |
0.029 |
170 |
312 |
|
(12.99) |
(14.41) |
(3.43) |
(8.12) |
(15.99) |
(-4.61) |
(6.18) |
||||
|
Singapore |
0.12 |
0.09 |
0.06 |
0.11 |
0.08 |
-0.01 |
0.10 |
0.066 |
171 |
669 |
|
(28.46) |
(18.07) |
(14.29) |
(22.76) |
(11.11) |
(-0.93) |
(16.53) |
||||
|
Mexico |
0.03 |
0.04 |
0.02 |
0.02 |
0.04 |
-0.01 |
0.01 |
0.085 |
171 |
116 |
|
(14.20) |
(14.68) |
(0.00) |
(8.31) |
(13.61) |
(-4.12) |
(0.00) |
||||
|
Norway |
0.05 |
0.01 |
0.02 |
0.07 |
0.02 |
0.01 |
0.07 |
0.027 |
171 |
240 |
|
(12.57) |
(3.63) |
(4.13) |
(14.28) |
(2.78) |
(1.76) |
(15.21) |
||||
|
Indonesia |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.062 |
165 |
428 |
|
(21.75) |
(20.17) |
(18.42) |
(11.93) |
(6.36) |
(5.29) |
(9.60) |
||||
|
Denmark |
0.10 |
0.08 |
0.05 |
0.09 |
0.08 |
-0.02 |
0.09 |
0.112 |
170 |
168 |
|
(11.19) |
(9.61) |
(7.31) |
(13.14) |
(9.47) |
(-2.89) |
(15.12) |
||||
|
Thailand |
0.01 |
0.01 |
0.00 |
0.01 |
0.01 |
-0.01 |
0.01 |
0.033 |
166 |
564 |
|
(11.34) |
(13.69) |
(5.26) |
(7.96) |
(14.37) |
(-8.31) |
(6.56) |
||||
|
Finland |
0.20 |
0.17 |
0.09 |
0.20 |
0.16 |
-0.04 |
0.17 |
0.142 |
171 |
126 |
|
(34.89) |
(23.79) |
(10.51) |
(43.00) |
(13.33) |
(-3.98) |
(34.40) |
||||
|
Turkey |
0.07 |
0.05 |
0.01 |
0.09 |
0.03 |
-0.01 |
0.08 |
0.020 |
171 |
317 |
|
(9.94) |
(8.75) |
(1.14) |
(14.78) |
(7.45) |
(-1.23) |
(12.20) |
||||
|
Poland |
0.09 |
0.06 |
0.01 |
0.12 |
0.05 |
-0.01 |
0.11 |
0.000 |
170 |
437 |
|
(15.64) |
(8.21) |
(2.58) |
(29.70) |
(5.74) |
(-1.83) |
(27.61) |
||||
|
Colombia |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.058 |
166 |
47 |
|
(8.77) |
(10.28) |
(5.30) |
(7.42) |
(9.65) |
(-5.60) |
(1.37) |
||||
|
Austria |
0.14 |
0.12 |
0.05 |
0.15 |
0.14 |
-0.06 |
0.13 |
0.076 |
169 |
75 |
|
(8.29) |
(5.42) |
(4.34) |
(8.34) |
(4.53) |
(-3.05) |
(9.59) |
||||
|
Philippines |
0.00 |
-0.03 |
-0.01 |
0.05 |
-0.04 |
0.01 |
0.05 |
0.000 |
166 |
228 |
|
(1.11) |
(-7.22) |
(-4.07) |
(9.05) |
(-7.86) |
(3.61) |
(9.52) |
||||
|
Argentina |
0.06 |
0.07 |
0.04 |
0.04 |
0.07 |
-0.02 |
0.02 |
0.037 |
164 |
78 |
|
(10.22) |
(11.22) |
(5.48) |
(5.97) |
(7.51) |
(-2.02) |
(3.15) |
||||
|
Ireland |
0.03 |
0.05 |
0.02 |
0.00 |
0.11 |
-0.06 |
-0.02 |
0.000 |
173 |
32 |
|
(1.86) |
(2.43) |
(1.27) |
(-0.00) |
(4.05) |
(-2.80) |
(-0.56) |
||||
|
Greece |
0.19 |
0.16 |
0.08 |
0.19 |
0.12 |
-0.01 |
0.14 |
0.108 |
167 |
238 |
|
(20.64) |
(17.34) |
(8.56) |
(31.41) |
(10.97) |
(-1.11) |
(30.07) |
||||
|
Portugal |
0.15 |
0.15 |
0.09 |
0.09 |
0.15 |
-0.02 |
0.06 |
0.047 |
173 |
50 |
|
(11.06) |
(9.99) |
(5.72) |
(5.24) |
(7.12) |
(-0.85) |
(3.75) |
||||
|
Peru |
0.11 |
0.10 |
0.06 |
0.10 |
0.07 |
0.00 |
0.07 |
0.103 |
171 |
120 |
|
(19.47) |
(15.70) |
(10.20) |
(19.00) |
(7.54) |
(-0.40) |
(10.66) |
||||
|
Czech Republic |
0.01 |
0.02 |
0.01 |
0.00 |
0.02 |
0.00 |
0.00 |
0.000 |
171 |
19 |
|
(5.70) |
(6.73) |
(6.08) |
(-0.30) |
(4.13) |
(-0.38) |
(-0.92) |
||||
|
New Zealand |
0.12 |
0.02 |
0.06 |
0.16 |
0.02 |
0.04 |
0.16 |
0.043 |
171 |
122 |
|
(13.74) |
(2.08) |
(6.01) |
(11.25) |
(1.55) |
(4.78) |
(10.34) |
||||
|
Chile |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.001 |
170 |
190 |
|
(6.33) |
(-1.97) |
(-2.43) |
(11.43) |
(-1.62) |
(-1.39) |
(9.64) |
||||
|
Pakistan |
0.01 |
0.01 |
0.01 |
0.01 |
0.01 |
0.00 |
0.01 |
0.070 |
167 |
246 |
|
(20.33) |
(14.38) |
(13.22) |
(23.94) |
(7.74) |
(0.31) |
(23.69) |
||||
In the analysis
on prices above we found that the price of stocks varies monotonically with
quality. We now perform a similar portfolio analysis on excess return. Although
higher quality firms command higher prices, higher quality should not provide
any excess return. We use the method of section 4.4 to create the 10 quality-sorted
portfolios. The results of the different regressions are reported in Table A-2
of Appendix 1. Because of computer memory restrictions I had to perform
regressions on one country sample at a time and it is thus not possible to
infer statistical significance of a global sample.
First, I calculated
the value-weighted excess return of the portfolios averaged over time and the Fama-Macbeth
method for estimates and standard errors. Asness et al. do not have any lag
between the measured returns and time of portfolio formation, i.e. they make no
assumption on the hypothetical investor’s holding period. Because we have
verified the persistence of quality, I have not included a sensitivity study on
the time lag between measured return and portfolio formation.
Secondly, I
have performed regressions of the portfolio excess returns using three models:
the CAPM market factor, the Fama-French 3-factor model and the
Fama-French-Carhart 4-factor model. For each regression I have reported the regression
constant alpha which is the
average excess return that is not due to sensitivity to the risk factors in the
model. The t-statistic of the alpha tells us whether the portfolio generates
statistically significant average abnormal returns. We also report the adjusted
R2 for the 4-factor model regression, the Sharpe ratio of the
portfolio’s excess returns and the information ratio for the 4-factor alpha – defined
as the annualized alpha divided by its annualized standard deviation and which
can be interpreted as the Sharpe ratio adjusted for hedging the four factor
exposures.
The excess return
for each portfolio is calculated using the Fama-Macbeth procedure by regressing the time series means
on a constant and implementing the Newey-West adjusted standard errors to
correct for autocorrelation. To obtain the alpha estimates on the CAPM,
3-factor and 4-factor specifications I run regressions of excess return on the
factor combinations and report the intercept after controlling for the CAPM,
Frama-French
Where the
factor-mimicking portfolios are made up as follows:
Because the full
panel data results in Table A-2 is quite big I have summarized the main
findings into charts and included a short excerpt as Table 4‑3. From Figure 4‑10 we see the global average of mean portfolio excess
returns for all countries which exhibit statistically significant difference
portfolios(changing to an equal weight of country results does not change the
interpretation). The very clear finding from the portfolio analysis confirms
our research hypothesis that investing in high quality companies and/or
shorting low quality earns a significantly positive excess return.
The difference
portfolio earns an excess of 0.98% per month (12.4% annualized) and the results
are monotonically increasing. Of our 44 countries we see statistically
significant result in 21 countries, accounting for 64% of the global market
capitalization. When controlling for market risk and the other common risk
factors we see that the monotonicity across portfolios remain, but the abnormal
excess returns of the difference portfolio is reduced from 98 to 70 basis
points per month.
Figure 4‑11 shows the 4-factor alpha for our
entire sample, split into those with statistically significant estimates and
not. The number of countries with significant alpha estimates for the
difference portfolio has decreased to 16 countries so we can only draw
inference for those shown in blue in the chart. Of the countries with
significant alphas the abnormal return varies from 68 to 242 basis points per
month, which is a very large magnitude.
The Sharpe
ratio is a performance measure of risk-adjusted return and is calculated as the
excess return divided by its standard deviation. Table A-2, exemplified in Table 4‑3, show that this ratio increases
with quality, The information
ratio (IR) is usually a measure of portfolio returns in excess of some
benchmark divided by its standard deviation, but Asness et al. calculate the IR
as the 4-factor alpha divided by the standard deviation of residuals. Figure 4‑12 shows the global market size weighted mean values for
Sharpe ratio and information ratio across the quality sorted portfolios. The chart
tells us that not only do quality stocks provide a higher return, but they are
also safer in the meaning
that we get higher returns per unit risk when investing in quality stocks over
“junky” stocks.
The results are
in line with what Asness et al. finds in their table 3 and it supports our
research hypothesis 2 – that there is a positive risk-adjusted return from investing in
high-quality stocks and shorting the low-quality stocks. Asness et al. use
these findings to support their argument that limited market efficiency
explains why quality only explain asset prices to a limited extent. They argue
that if high quality stocks earn a higher risk-adjusted return than low quality
stocks it must imply that market prices fail to reflect the quality
characteristics. Alternatively, quality is linked to risk in a way which is not
fully captured by the safety sub-factor of quality. We will explore this in the
next section.
Figure 4‑10: The chart shows market-weighted means of the countries with statistically significant difference portfolios for excess returns and 4-factor alphas.
The blue bars represent the global average
excess return of the quality-sorted portfolios along with the difference
portfolios high-low. The peach bars represent the global average abnormal alpha
returns when regressing portfolio excess returns on the 4 Fama-French-Carhart
factors.
Figure 4‑11: This chart shows the fama-french-carhart abnormal returns (alpha) for the regressions of the quality-sorted difference portfolio’s monthly excess return
on the Fama-French-Carhart four factors. The
countries is blue have statistically significant alpha estimates, whilst the
countries in peach are included only for reference as their estimates are not
significantly different from zero.
Table 4‑3: Return on Quality - excerpt from Table A-2 showing two countries of interest
The table shows quality sorted portfolio excess monthly returns along with the
difference portfolio high-low. The alphas are the intercept from time series
regressions of the monthly excess returns using the CAPM, Fama-French 3-factor
model and the Fama-French 4-factor regression models. Returns and alphas in monthly percentage,
t-statistics are shown below estimates in paranthesis and 5% statistical
significance is shown in bold. Sharpe ratios (run on excess returns) and
information ratios (run on 4-factor alphas) are annualized.
|
|
P1 |
P2 |
P3 |
P4 |
P5 |
P6 |
P7 |
P8 |
P9 |
P10 |
H-L |
|
Panel A: US Sample |
|||||||||||
|
Excess Return |
-1.86 |
-1.13 |
-0.88 |
-0.90 |
0.33 |
-0.60 |
-0.22 |
-0.39 |
-0.20 |
-0.11 |
1.75 |
|
|
(-1.73) |
(-1.14) |
(-0.97) |
(-0.99) |
(0.26) |
(-0.74) |
(-0.26) |
(-0.50) |
(-0.26) |
(-0.14) |
(3.68) |
|
CAPM α |
-1.18 |
-0.76 |
-0.29 |
-0.32 |
0.03 |
-0.07 |
0.19 |
0.06 |
-0.10 |
0.38 |
1.55 |
|
|
(-2.36) |
(-1.31) |
(-0.69) |
(-0.74) |
(0.04) |
(-0.20) |
(0.54) |
(0.16) |
(-0.36) |
(1.45) |
(3.34) |
|
3-factor α |
-1.06 |
-0.25 |
-0.48 |
-0.26 |
-0.48 |
-0.07 |
-0.01 |
0.02 |
-0.19 |
0.08 |
1.14 |
|
|
(-2.55) |
(-0.49) |
(-1.13) |
(-0.67) |
(-0.97) |
(-0.24) |
(-0.03) |
(0.04) |
(-0.58) |
(0.27) |
(3.60) |
|
4-factor α |
-0.95 |
-0.68 |
-0.52 |
0.00 |
-0.41 |
-0.07 |
-0.15 |
-0.12 |
-0.31 |
-0.18 |
0.77 |
|
|
(-2.57) |
(-1.08) |
(-1.37) |
(-0.02) |
(-0.94) |
(-0.22) |
(-0.42) |
(-0.39) |
(-0.91) |
(-0.67) |
(3.23) |
|
Sharpe Ratio |
-0.63 |
-0.43 |
-0.36 |
-0.37 |
0.11 |
-0.27 |
-0.10 |
-0.19 |
-0.10 |
-0.05 |
1.96 |
|
Information Ratio |
-1.14 |
-0.75 |
-0.77 |
-0.01 |
-0.55 |
-0.10 |
-0.21 |
-0.16 |
-0.42 |
-0.30 |
1.04 |
|
Panel B: Norwegian Sample |
|||||||||||
|
Excess Return |
-0.35 |
-0.51 |
0.34 |
0.25 |
0.22 |
0.33 |
0.70 |
0.64 |
0.67 |
0.92 |
1.27 |
|
(-0.54) |
(-0.83) |
(0.58) |
(0.46) |
(0.41) |
(0.72) |
(1.34) |
(1.46) |
(1.48) |
(1.89) |
(3.38) |
|
|
CAPM α |
-1.45 |
-3.84 |
0.87 |
0.64 |
0.57 |
0.47 |
0.84 |
1.17 |
0.52 |
1.31 |
2.76 |
|
(-1.24) |
(-1.35) |
(1.71) |
(1.22) |
(1.42) |
(1.49) |
(2.21) |
(2.60) |
(0.72) |
(3.54) |
(2.28) |
|
|
3-factor α |
-0.36 |
-0.39 |
0.90 |
0.36 |
1.04 |
0.77 |
1.02 |
0.85 |
0.96 |
1.32 |
1.68 |
|
(-0.59) |
(-0.84) |
(1.96) |
(0.77) |
(2.77) |
(2.28) |
(2.86) |
(1.94) |
(2.22) |
(3.20) |
(2.38) |
|
|
4-factor α |
-0.64 |
-0.41 |
0.86 |
0.19 |
0.97 |
0.39 |
1.00 |
0.85 |
0.82 |
1.25 |
1.89 |
|
(-1.11) |
(-0.81) |
(1.71) |
(0.42) |
(2.64) |
(1.04) |
(2.45) |
(2.00) |
(1.82) |
(3.42) |
(2.87) |
|
|
Sharpe Ratio |
-0.18 |
-0.28 |
0.22 |
0.16 |
0.16 |
0.26 |
0.48 |
0.49 |
0.50 |
0.72 |
0.93 |
|
Information Ratio |
-0.26 |
-0.21 |
0.50 |
0.12 |
0.75 |
0.31 |
0.64 |
0.54 |
0.51 |
0.95 |
0.73 |
Figure 4‑12: This chart shows the global market weight mean of the Sharpe ratios and the information ratios of quality-sorted portfolios
along
with the difference portfolio (High-Low). The Sharpe Ratio is calculated on the
monthly mean excess return of the portfolio and the information ratio is
calculated on the intercept (alpha) of regressions of portfolio returns on the
Fama-French-Carhart four factors.
We now create
the QMJ (quality minus junk) factor in the same way as Asness et al. The QMJ
factor is created by taking the value weighted returns of the intersection of
six portfolios formed on size and quality. Each month we form two portfolios,
small and large, based on the available asset’s market capitalization. In their
study they use a different breakpoint for US and global stocks, but I have used
the 80th percentile of market equity as breakpoint for all countries.
Quality is sorted into ten portfolios and we define the top 30% as quality
stocks and the bottom 30% as junk. QMJ then becomes:
With the factor
returns we run a regression of QMJ on the Fama-French-Carhart risk factors and
report the alpha from regressions on the CAPM model and Fama-French 3-factor
model. Regressions are run for each time period and standard errors corrected
as per the Fama-Macbeth procedure and corrected for autocorrelation using the
Newey-West method.
The results are
shown in Table 4‑4 for the full sample of countries. The key takeaway is
that the QMJ factor delivers significant excess return and alpha with respect
to the various risk factors. The alpha is the average excess return that is not
due to sensitivity to the risk factors in the model. The t-statistic of the
alpha tells us whether the QMJ portfolio generates statistically significant
average abnormal returns. The excess monthly return is on average 0.24% (2.9% per
year) among the countries with significant estimates and the abnormal returns
(alpha) averaging at 0.31% and significantly non-zero for 32 of the 44
countries.
From the risk-factor
loadings on MKT, SMB, HML and MOM we see that quality has significant negative
exposures to all except the momentum factor (MOM). The negative exposure to market
(MKT) and size (SMB) tells us that QMJ is long on low-beta and large stocks
and/or short high-beta small cap stocks. QMJ is negatively loaded on the value
factor (HML) which can be explained because quality is positively related to
price whilst HML is long on cheap stocks. The exposure to momentum is small and
not significant, but I kept it in the model for comparison with Asness et al. The
Sharpe ratio is calculated on the excess returns, but the more interesting
comparison is the information ratio which can be interpreted as the Sharpe
ratio adjusted for hedging the other factor exposures. Figure 4‑13 shows us that across our entire sample the QMJ factor
delivers a positive alpha (except Ireland) and information ratio, even
considering our short time series of data.
Generally, the
results on the QMJ factor match very well with what Asness et al.’s study on 24
countries found. A QMJ portfolio that is long high-quality and short junk
stocks earn a large and significant abnormal return (alpha) when controlling
for some of the most used risk-factors used in literature. From the factor
loadings the QMJ portfolio appear safer and this is also supported by the high
information ratio (Sharpe ratio after hedging for other factor exposures) above
1.
Figure 4‑13: QMJ 4-factor alpha information ratios.
This chart plots the Fama-French-Carhart
adjusted information ratio of the Quality minus Junk (QMJ) factor.
Table 4‑4: Regressions of QMJ returns on risk factors.
This table shows the QMJ portfolio returns and factor loadings on the
Fama-French-Carhart risk factors along with the intercept (alphas) of
time-series regressions of monthly returns on the CAPM and Fama-French 3-factor
model. Returns and alphas are monthly percentages and t-statistics are shown in
parentheses under the coefficient estimates.Annualized Sharpe ratio is
calculated on the QMJ portfolio excess return and the information ratio on the
4-factor alpha.
|
QMJ |
Excess
Return |
CAPM
α |
3-factor
α |
4-factor
α |
MKT |
SMB |
HML |
MOM |
Sharpe
Ratio |
IR |
Adj.
R2 |
|
Sample Mean |
0.24 |
0.30 |
0.33 |
0.31 |
-0.06 |
-0.10 |
-0.16 |
0.09 |
0.67 |
1.06 |
0.38 |
|
United States |
0.25 |
0.38 |
0.33 |
0.32 |
-0.10 |
-0.19 |
-0.27 |
0.04 |
0.58 |
1.39 |
0.71 |
|
(1.97) |
(3.95) |
(4.75) |
(4.23) |
(-5.77) |
(-6.01) |
(-4.74) |
(0.70) |
|
|
|
|
|
Great Britain |
0.64 |
0.66 |
0.71 |
0.62 |
-0.03 |
-0.01 |
-0.17 |
0.19 |
2.13 |
2.52 |
0.31 |
|
(7.27) |
(8.95) |
(8.46) |
(6.51) |
(-0.99) |
(-0.12) |
(-5.07) |
(3.04) |
||||
|
Japan |
0.07 |
0.05 |
0.18 |
0.18 |
0.01 |
0.05 |
-0.32 |
-0.10 |
0.21 |
0.97 |
0.59 |
|
(0.84) |
(0.68) |
(3.34) |
(3.28) |
(0.78) |
(2.05) |
(-12.52) |
(-2.68) |
||||
|
Hong Kong |
0.04 |
0.08 |
0.15 |
0.16 |
-0.08 |
-0.06 |
-0.16 |
-0.01 |
0.09 |
0.46 |
0.23 |
|
(0.29) |
(0.76) |
(1.49) |
(1.52) |
(-3.35) |
(-3.39) |
(-3.38) |
(-0.14) |
||||
|
China |
0.13 |
0.22 |
0.36 |
0.40 |
-0.03 |
-0.08 |
-0.29 |
0.20 |
0.24 |
1.14 |
0.56 |
|
(1.00) |
(1.99) |
(3.21) |
(3.82) |
(-2.46) |
(-2.29) |
(-6.90) |
(2.23) |
||||
|
Canada |
0.16 |
0.25 |
0.70 |
0.77 |
-0.17 |
-0.19 |
0.05 |
0.06 |
0.30 |
1.98 |
0.40 |
|
(0.86) |
(1.72) |
(4.50) |
(4.21) |
(-5.09) |
(-4.79) |
(0.65) |
(0.82) |
||||
|
Germany |
0.42 |
0.44 |
0.54 |
0.52 |
-0.12 |
-0.23 |
-0.07 |
0.05 |
1.28 |
1.77 |
0.25 |
|
(4.52) |
(4.72) |
(6.09) |
(5.57) |
(-4.10) |
(-5.56) |
(-1.73) |
(1.04) |
||||
|
Australia |
0.38 |
0.42 |
0.55 |
0.56 |
-0.12 |
-0.17 |
0.12 |
0.03 |
1.05 |
1.99 |
0.37 |
|
(3.61) |
(4.49) |
(0.00) |
(6.85) |
(-5.94) |
(-5.70) |
(4.11) |
(0.78) |
||||
|
South Korea |
0.15 |
0.21 |
0.33 |
0.32 |
-0.06 |
-0.01 |
-0.17 |
0.07 |
0.34 |
0.81 |
0.17 |
|
(1.16) |
(1.68) |
(2.55) |
(2.54) |
(-2.65) |
(-0.37) |
(-3.67) |
(1.00) |
||||
|
Switzerland |
0.35 |
0.33 |
0.31 |
0.32 |
0.03 |
-0.04 |
-0.14 |
-0.02 |
1.02 |
0.98 |
0.08 |
|
(4.91) |
(4.56) |
(4.53) |
(4.75) |
(0.92) |
(-0.88) |
(-3.14) |
(-0.36) |
||||
|
Switzerland |
0.35 |
0.33 |
0.31 |
0.32 |
0.03 |
-0.04 |
-0.14 |
-0.02 |
1.02 |
0.98 |
0.08 |
|
(4.91) |
(4.56) |
(4.53) |
(4.75) |
(0.92) |
(-0.88) |
(-3.14) |
(-0.36) |
||||
|
Russia |
0.22 |
0.29 |
0.36 |
0.38 |
-0.08 |
-0.01 |
0.13 |
0.31 |
0.24 |
0.45 |
0.14 |
|
(0.70) |
(0.94) |
(1.44) |
(1.68) |
(-1.79) |
(-0.10) |
(2.35) |
(2.17) |
||||
|
Spain |
0.46 |
0.56 |
0.55 |
0.57 |
-0.20 |
-0.24 |
-0.26 |
-0.03 |
0.80 |
1.29 |
0.40 |
|
(2.40) |
(3.27) |
(3.70) |
(3.91) |
(-4.96) |
(-4.67) |
(-5.90) |
(-0.43) |
||||
|
Spain |
0.46 |
0.56 |
0.55 |
0.57 |
-0.20 |
-0.24 |
-0.26 |
-0.03 |
0.80 |
1.29 |
0.40 |
|
(2.40) |
(3.27) |
(3.70) |
(3.91) |
(-4.96) |
(-4.67) |
(-5.90) |
(-0.43) |
||||
|
Sweden |
0.51 |
0.56 |
0.49 |
0.38 |
-0.09 |
-0.07 |
-0.02 |
0.24 |
1.19 |
1.00 |
0.18 |
|
(4.78) |
(5.36) |
(4.76) |
(3.29) |
(-3.60) |
(-1.65) |
(-0.41) |
(3.44) |
||||
|
Malaysia |
0.57 |
0.62 |
0.63 |
0.57 |
-0.09 |
-0.12 |
-0.40 |
0.18 |
0.94 |
1.54 |
0.63 |
|
(3.84) |
(4.51) |
(5.47) |
(4.16) |
(-3.69) |
(-4.31) |
(-10.15) |
(2.60) |
||||
|
Malaysia |
0.30 |
0.40 |
0.48 |
0.46 |
-0.08 |
-0.04 |
-0.21 |
0.10 |
0.82 |
1.60 |
0.35 |
|
(2.83) |
(4.37) |
(4.62) |
(4.66) |
(-2.21) |
(-1.17) |
(-3.87) |
(1.46) |
||||
|
Taiwan |
0.18 |
0.17 |
0.40 |
0.46 |
0.03 |
-0.05 |
-0.45 |
-0.14 |
0.40 |
1.56 |
0.54 |
|
(1.61) |
(1.53) |
(4.29) |
(4.29) |
(1.88) |
(-1.53) |
(-9.84) |
(-2.07) |
||||
|
Brazil |
0.25 |
0.37 |
0.60 |
0.61 |
-0.09 |
-0.07 |
-0.09 |
-0.02 |
0.33 |
0.94 |
0.04 |
|
(1.37) |
(1.95) |
(2.51) |
(2.42) |
(-2.06) |
(-1.11) |
(-0.98) |
(-0.20) |
||||
|
Netherlands |
0.37 |
0.48 |
0.43 |
0.31 |
-0.12 |
-0.08 |
-0.12 |
0.19 |
0.58 |
0.56 |
0.27 |
|
(2.34) |
(3.05) |
(2.82) |
(2.01) |
(-2.87) |
(-1.72) |
(-2.64) |
(2.18) |
||||
|
South Africa |
0.30 |
0.33 |
0.44 |
0.43 |
-0.08 |
-0.09 |
-0.17 |
0.12 |
0.71 |
1.15 |
0.16 |
|
(2.48) |
(2.74) |
(4.32) |
(4.13) |
(-2.36) |
(-2.68) |
(-4.80) |
(1.48) |
||||
|
Singapore |
0.25 |
0.31 |
0.43 |
0.46 |
-0.09 |
-0.18 |
-0.01 |
0.13 |
0.50 |
1.16 |
0.33 |
|
(1.81) |
(2.40) |
(3.53) |
(3.56) |
(-2.95) |
(-5.13) |
(-0.27) |
(2.37) |
||||
|
Mexico |
0.25 |
0.26 |
0.36 |
0.26 |
0.00 |
-0.05 |
-0.16 |
0.15 |
0.48 |
0.52 |
0.11 |
|
(1.79) |
(1.72) |
(2.64) |
(1.73) |
(-0.06) |
(-0.77) |
(-2.83) |
(2.00) |
||||
|
Norway |
0.43 |
0.48 |
0.43 |
0.38 |
-0.14 |
-0.15 |
-0.10 |
0.08 |
0.73 |
0.70 |
0.12 |
|
(2.29) |
(2.74) |
(2.66) |
(2.40) |
(-3.60) |
(-1.96) |
(-2.10) |
(1.02) |
||||
|
Indonesia |
0.17 |
0.15 |
0.33 |
0.33 |
-0.03 |
-0.13 |
-0.03 |
0.03 |
0.30 |
0.59 |
0.02 |
|
(0.94) |
(0.78) |
(1.97) |
(1.96) |
(-0.58) |
(-2.13) |
(-0.46) |
(0.43) |
||||
|
Denmark |
0.46 |
0.55 |
0.37 |
0.21 |
-0.05 |
-0.11 |
-0.26 |
0.26 |
0.72 |
0.39 |
0.27 |
|
(1.86) |
(2.34) |
(2.02) |
(1.18) |
(-1.46) |
(-1.82) |
(-3.55) |
(3.74) |
||||
|
Thailand |
0.03 |
0.15 |
0.26 |
0.29 |
-0.16 |
-0.13 |
-0.18 |
-0.05 |
0.09 |
0.95 |
0.33 |
|
(0.30) |
(1.47) |
(2.67) |
(2.71) |
(-6.00) |
(-3.86) |
(-4.71) |
(-1.11) |
||||
|
Finland |
0.44 |
0.44 |
0.43 |
0.26 |
0.04 |
0.02 |
-0.13 |
0.25 |
0.88 |
0.55 |
0.12 |
|
(3.14) |
(3.18) |
(3.35) |
(1.80) |
(1.01) |
(0.33) |
(-2.33) |
(2.49) |
||||
|
Turkey |
0.28 |
0.31 |
0.38 |
0.29 |
0.00 |
-0.08 |
-0.02 |
0.26 |
0.56 |
0.64 |
0.13 |
|
(2.14) |
(2.48) |
(3.41) |
(2.53) |
(-0.14) |
(-1.46) |
(-0.42) |
(2.58) |
||||
|
Chile |
0.25 |
0.36 |
0.42 |
0.34 |
-0.12 |
0.02 |
-0.08 |
0.13 |
0.61 |
0.89 |
0.13 |
|
(2.59) |
(4.29) |
(4.31) |
(2.95) |
(-1.90) |
(0.30) |
(-1.47) |
(1.55) |
||||
|
Poland |
0.19 |
0.29 |
0.33 |
0.25 |
-0.12 |
-0.07 |
-0.04 |
0.18 |
0.34 |
0.53 |
0.25 |
|
(1.08) |
(1.72) |
(2.06) |
(1.57) |
(-3.23) |
(-2.24) |
(-0.68) |
(2.87) |
||||
|
Colombia |
0.24 |
0.13 |
0.18 |
0.15 |
0.07 |
-0.05 |
0.02 |
0.18 |
0.22 |
0.14 |
0.02 |
|
(0.89) |
(0.47) |
(0.82) |
(0.70) |
(0.67) |
(-0.32) |
(0.15) |
(0.86) |
||||
|
Austria |
0.15 |
0.20 |
0.34 |
0.29 |
-0.11 |
-0.11 |
-0.14 |
0.27 |
0.21 |
0.49 |
0.32 |
|
(0.82) |
(1.27) |
(2.23) |
(2.08) |
(-2.78) |
(-2.13) |
(-2.72) |
(4.53) |
||||
|
Philippines |
0.14 |
0.27 |
0.48 |
0.48 |
-0.12 |
-0.11 |
-0.19 |
0.00 |
0.18 |
0.70 |
0.10 |
|
(0.66) |
(1.37) |
(2.35) |
(2.34) |
(-2.68) |
(-1.66) |
(-3.13) |
(-0.03) |
||||
|
Argentina |
-0.02 |
0.33 |
0.35 |
0.25 |
-0.12 |
0.05 |
-0.07 |
0.33 |
-0.02 |
0.25 |
0.17 |
|
(-0.08) |
(1.52) |
(1.56) |
(1.19) |
(-3.35) |
(0.80) |
(-1.35) |
(2.42) |
||||
|
Ireland |
-0.46 |
-0.47 |
-0.42 |
-0.49 |
0.01 |
-0.07 |
-0.06 |
0.28 |
-0.28 |
-0.31 |
0.05 |
|
(-1.01) |
(-1.03) |
(-1.01) |
(-1.19) |
(0.09) |
(-0.66) |
(-0.88) |
(2.39) |
||||
|
Greece |
0.56 |
0.50 |
0.77 |
0.89 |
-0.09 |
-0.07 |
-0.26 |
0.14 |
0.60 |
1.33 |
0.46 |
|
(2.35) |
(2.40) |
(4.20) |
(4.51) |
(-1.85) |
(-1.27) |
(-4.87) |
(1.87) |
||||
|
Portugal |
0.47 |
0.54 |
0.40 |
0.54 |
-0.20 |
0.09 |
0.12 |
0.33 |
0.41 |
0.53 |
0.20 |
|
(1.53) |
(1.98) |
(1.45) |
(1.85) |
(-2.43) |
(1.46) |
(1.87) |
(3.83) |
||||
|
Peru |
0.26 |
0.14 |
0.42 |
0.42 |
0.18 |
0.03 |
-0.30 |
0.00 |
0.28 |
0.52 |
0.17 |
|
(1.07) |
(0.69) |
(2.13) |
(2.13) |
(1.51) |
(0.34) |
(-3.94) |
(-0.01) |
||||
|
Czech
Republic |
0.26 |
0.34 |
0.44 |
0.51 |
-0.03 |
0.07 |
-0.06 |
0.37 |
0.25 |
0.52 |
0.10 |
|
(0.83) |
(1.08) |
(1.39) |
(1.64) |
(-0.40) |
(0.81) |
(-0.87) |
(3.06) |
||||
|
New Zealand |
0.24 |
0.31 |
0.33 |
0.35 |
-0.18 |
-0.14 |
0.05 |
0.21 |
0.36 |
0.58 |
0.14 |
|
(1.42) |
(1.68) |
(1.80) |
(1.89) |
(-3.06) |
(-3.01) |
(1.24) |
(2.26) |
||||
|
Pakistan |
0.40 |
0.42 |
0.64 |
0.63 |
-0.04 |
-0.19 |
-0.14 |
0.02 |
0.57 |
1.02 |
0.17 |
|
(1.87) |
(2.02) |
(3.47) |
(3.48) |
(-1.46) |
(-3.46) |
(-1.94) |
(0.20) |
I also look
at the intertemporal properties to see what they tell us about how quality
varies over time. Although the sample data only stretches from 2005 to 2019 it
does include very shifting economic periods. The boom time leading up to the
financial crisis in 2008 is one environment and the following downturn lasting
until 2009 another. And finally, the ten-year bull rally up until 2019. From Figure 4‑14 we see the average cumulative
factor returns for the global sample. The market ups and down can be seen on
the black line and we also see how quality on average performs very stable
during the time sample. Asness et al. describe a “flight to quality”, i.e. when
the market goes down investors flock to high quality stocks and drive their
returns up. The same phenomenon is proven for the value factor and we see that
the HML returns heavily influenced by high returns during the financial crisis.
My data does not show an equally strong effect on QMJ, but we do see that the
QMJ returns are in fact positive during the financial crisis. From Figure 4‑15 we can see the time series of the
price of quality (which we analyzed in section 4.5.2). The chart shows the that the
slope coefficient of quality in regressions of price on quality increase by
100% during the financial crisis compared to the pre-crisis values.
The “slow
and steady” performance of a long quality, sort junk strategy is also shown in Figure 4‑16. It displays, for the US sample,
the cumulative abnormal returns using the 4-factor alpha for the quality sorted
difference (H-L) portfolio. We see that through bull and bear markets the
investment in high quality stocks yield an almost perfectly linear return when
hedging for the other factors. The figure is very much in line with the equivalent
figures in Asness et al.
As a
curiosity, I have also included the cumulative 4-factor alpha of the quality
sorted portfolio for the Norwegian sample in Figure 4‑17. Here we se a clear shift in
abnormal returns after the financial crisis, with quality (H-L) earning huge
returns after hedging for other risk factors.
In
conclusion, we observe the same time-varying properties of quality as found by
Asness et al. QMJ and the quality difference portfolio deliver consistent
positive risk-adjusted returns over time and it looks to be robust to varying
economic environments.
Figure 4‑14: The chart shows the Global average cumulative excess returns from the factor portfolios
formed each month for the factors MKT (market),
HML (Value), QMJ (Quality), MOM (Momentum) and SMB (size).
Figure 4‑15: The time-varying price of quality.
This chart shows the slope coefficients from
monthly cross-sectional regressions of price on quality as defined in section 4.5.2. The sample is US stocks.
Figure 4‑16: This chart shows the cumulative 4-factor alpha factor abnormal returns
for the difference portfolio (high-low) sorted
on quality. Sample is the US Stock Market.
Figure 4‑17: Abnormal returns (alpha) of the Quality sorted portfolios when adjusted for Fama-French-Carhart risk factors.
Sample is norwegian stocks and the estimates
are significant for all portfolios from 6 and above.
Figure 4‑18: Cumulative excess returns of a QMJ portfolio formed every month.
Each line represents one of the 44 countries
in the global sample. Ireland is the only loss-making country.
In this
section I explain how the QMJ factor can be used in modern investment practice.
With the rise of computing power and accessible data over the last decades, we
have also seen a rapid development in the vaguely defined field of quantitative
finance. This involves the use of mathematical models on large datasets of
financial and can refer to algorithmic trading methods, data driven research
and analysis or other way of applying math to draw conclusions from datasets.
The workflow of developing a quantitative investment strategy, shown in Figure 4‑19, can be explained by breaking it
into several models
The return
model, somethings called the alpha model, is the heart of the strategy. This
model will build on some investment hypothesis that can be used to predict the
relative movements of future returns for the asset class being analyzed. For
example, we could hypothesize that the future return of a firm’s stock
correlates with how often it is mentioned on Twitter, by the historic stock
prices or some other pattern. The return model is then tested on the data and
its effectiveness can be analyzed by various statistical measures. In the
analysis work we have shown how the QMJ alpha looks to be robust and could form
the basis of a suitable return model.
The purpose
of the risk model is to enable constructing a portfolio in line with the
investors risk profile and to minimize risk of losses. This is done by
evaluating the performance of the constructed portfolio against some measure of
risk. The traditional measure used by Markowitz was portfolio return variance,
but more modern risk models use measures like the Sharpe ratio, exposure to
factors (e.g. style and sector) or portfolio drawdown to understand the
portfolio risk and set limits to exposure.
The
portfolio construction model combines and perform trade-off studies between the
return model and the risk model combined with a model to take into account
trading cost and constraints like the cost of buying a stock, slippage effects
or avoiding illiquid assets. Portfolio construction can be done using for
instance mean variance methods, but often the investor will optimize the
portfolio construction and execution model by performing back tests. Back
testing means that you run a simulation of how your algorithm would have
performed using historic data. The portfolio construction model describes the
rules for selecting assets and trade them at an acceptable cost and risk level
and the execution model simulates the actual trading on historic data.
Using this framework,
we could create a trading strategy based on the QMJ factor or include the QMJ
factor in an existing strategy to take advantage of the abnormal returns we
seem to achieve.
Figure 4‑19: Workflow for quantitative investment, adopted from
The goal of
this thesis is to examine whether the findings for Asness et al.’s paper “quality
minus junk”
The first
finding on the positive relationship between price and quality shows that the
model specification based on modern asset pricing has explanatory power on stock
prices, but most of the cross-sectional variation in prices is still unexplained.
This finding is the case in 40 out of the 44 countries examined for the sample
data between 2005 and 2019.
The second
finding shows that a factor-mimicking portfolio (QMJ) going long on the highest
quality firms and shorting the low-quality stocks earns a significant
risk-adjusted return with a Sharpe ratio after hedging for other factor
exposures just above 1. The risk-adjusted alpha was positive for 43 out of the
44 countries in the sample.
The third
finding of this study is that the abnormal quality returns are consistent
across the time period and appears robust to changing economic environments. The
price of quality stocks do increase during times of distress, indicating a
“flight to quality”, but the risk-adjusted alpha (when hedging for other risk
factors) does not fall during the financial crisis.
This thesis
confirms the results from Asness et al. using a broader sample of countries,
but with a short time horizon. In their paper they conclude the abnormal
returns of quality stock are due to mispricing and they are unable to find a
risk-based explanation. If anything, quality stocks are less risky than
lower-quality stocks as measured by Sharpe ratio. The other avenue to pursue
would be preference or behavior-based explanation to why the average investor
does not wish to hold quality stocks. I find that quality deliver consistent returns
during times of distress as well as in times of boom. It is therefore difficult
to argue that investors shy away from these stocks to avoid negative returns when
the marginal utility of consumption is high. It has been proposed that investor
bias like lottery preference and overconfidence
Recommendations
for further research would be to perform backtesting of a trading strategy
based on QMJ to test whether the strategy delivers abnormal returns also when
controlling for constraints like commissions, constraints on short trading,
etc. I would also dig deeper into the price of quality. We found quality to
explain very little of the cross-sectional variation in price, but by analyzing
the relationship between price and the individual accounting components that
make up the quality factor or regression of quality on other variables we might
learn more to understand why a theoretically sound valuation model explains so
little of cross-sectional variation in prices.
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Table A-1: Persistence of Quality measure
across quality sorted portfolios.
|
Country |
Time |
P1 |
P2 |
P3 |
P4 |
P5 |
P6 |
P7 |
P8 |
P9 |
P10 |
H-L |
t-stat |
|
Argentina |
t |
-1.75 |
-1.13 |
-0.69 |
-0.36 |
-0.08 |
0.19 |
0.46 |
0.71 |
1.05 |
1.63 |
3.37 |
(177.64) |
|
t + 3Y |
-0.68 |
-0.47 |
-0.45 |
-0.27 |
-0.11 |
0.06 |
0.13 |
0.13 |
0.45 |
0.70 |
1.38 |
(14.79) |
|
|
t + 10Y |
0.08 |
-0.12 |
-0.39 |
-0.38 |
-0.11 |
-0.17 |
-0.19 |
0.13 |
0.21 |
0.33 |
0.26 |
(1.27) |
|
|
Australia |
t |
-1.74 |
-1.05 |
-0.67 |
-0.37 |
-0.11 |
0.12 |
0.37 |
0.65 |
1.02 |
1.79 |
3.52 |
(335.83) |
|
t + 3Y |
-0.43 |
-0.29 |
-0.18 |
-0.13 |
-0.07 |
-0.01 |
0.14 |
0.29 |
0.42 |
0.73 |
1.15 |
(53.89) |
|
|
t + 10Y |
-0.36 |
-0.17 |
-0.05 |
-0.04 |
-0.01 |
0.07 |
0.11 |
0.18 |
0.23 |
0.52 |
0.88 |
(29.16) |
|
|
Austria |
t |
-1.61 |
-1.09 |
-0.76 |
-0.45 |
-0.15 |
0.15 |
0.45 |
0.76 |
1.07 |
1.67 |
3.28 |
(155.02) |
|
t + 3Y |
-0.69 |
-0.59 |
-0.27 |
-0.22 |
-0.17 |
0.29 |
0.10 |
0.38 |
0.59 |
0.55 |
1.24 |
(12.90) |
|
|
t + 10Y |
-0.65 |
-0.16 |
0.04 |
0.04 |
0.20 |
0.14 |
-0.11 |
0.47 |
0.16 |
-0.08 |
0.60 |
(5.97) |
|
|
Brazil |
t |
-1.66 |
-0.99 |
-0.67 |
-0.43 |
-0.20 |
0.04 |
0.33 |
0.66 |
1.10 |
1.82 |
3.48 |
(205.09) |
|
t + 3Y |
-0.76 |
-0.59 |
-0.50 |
-0.34 |
-0.26 |
-0.07 |
0.06 |
0.30 |
0.68 |
1.22 |
1.98 |
(39.08) |
|
|
t + 10Y |
-0.48 |
-0.44 |
-0.14 |
-0.12 |
-0.23 |
-0.20 |
-0.23 |
-0.01 |
0.24 |
0.53 |
0.99 |
(8.21) |
|
|
Canada |
t |
-1.83 |
-1.03 |
-0.63 |
-0.33 |
-0.09 |
0.15 |
0.39 |
0.65 |
1.00 |
1.72 |
3.55 |
(190.69) |
|
t + 3Y |
-0.61 |
-0.29 |
-0.16 |
-0.06 |
0.00 |
0.08 |
0.19 |
0.31 |
0.51 |
0.79 |
1.40 |
(23.90) |
|
|
t + 10Y |
-0.24 |
-0.05 |
0.00 |
-0.01 |
0.06 |
0.19 |
0.27 |
0.28 |
0.50 |
0.54 |
0.78 |
(9.26) |
|
|
Chile |
t |
-1.58 |
-0.98 |
-0.68 |
-0.43 |
-0.19 |
0.04 |
0.27 |
0.59 |
1.05 |
1.93 |
3.51 |
(291.95) |
|
t + 3Y |
-0.71 |
-0.50 |
-0.37 |
-0.24 |
-0.14 |
0.00 |
0.14 |
0.30 |
0.56 |
1.03 |
1.74 |
(26.93) |
|
|
t + 10Y |
-0.18 |
-0.27 |
-0.06 |
-0.13 |
-0.13 |
0.09 |
0.08 |
0.18 |
0.20 |
0.63 |
0.81 |
(5.61) |
|
|
China |
t |
-1.63 |
-1.06 |
-0.73 |
-0.44 |
-0.18 |
0.09 |
0.37 |
0.68 |
1.09 |
1.81 |
3.45 |
(506.64) |
|
t + 3Y |
-0.74 |
-0.67 |
-0.52 |
-0.40 |
-0.30 |
-0.16 |
-0.02 |
0.15 |
0.42 |
0.85 |
1.59 |
(57.92) |
|
|
t + 10Y |
-0.40 |
-0.39 |
-0.34 |
-0.38 |
-0.27 |
-0.23 |
-0.18 |
-0.22 |
-0.16 |
0.20 |
0.60 |
(11.61) |
|
|
Colombia |
t |
-1.76 |
-1.01 |
-0.60 |
-0.32 |
-0.09 |
0.14 |
0.41 |
0.68 |
1.03 |
1.64 |
3.40 |
(261.99) |
|
t + 3Y |
-0.99 |
-0.50 |
-0.51 |
-0.24 |
-0.19 |
-0.06 |
0.17 |
0.42 |
0.47 |
1.03 |
2.02 |
(13.21) |
|
|
t + 10Y |
-0.72 |
-0.21 |
-0.22 |
-0.21 |
-0.01 |
-0.45 |
-0.54 |
-0.37 |
-0.14 |
0.41 |
1.13 |
(5.68) |
|
|
Czech
Republic |
t |
-1.55 |
-1.02 |
-0.72 |
-0.41 |
-0.13 |
0.12 |
0.39 |
0.70 |
1.00 |
1.65 |
3.19 |
(78.37) |
|
t + 3Y |
-0.35 |
-0.17 |
-0.14 |
-0.29 |
-0.08 |
-0.01 |
0.37 |
0.31 |
0.19 |
0.28 |
0.64 |
(3.19) |
|
|
t + 10Y |
-1.20 |
-0.23 |
0.29 |
-0.60 |
0.48 |
0.65 |
0.09 |
0.09 |
-0.23 |
-0.06 |
1.82 |
(41.56) |
|
|
Denmark |
t |
-1.48 |
-0.96 |
-0.73 |
-0.53 |
-0.31 |
-0.02 |
0.31 |
0.67 |
1.14 |
1.92 |
3.40 |
(246.93) |
|
t + 3Y |
-0.29 |
-0.40 |
-0.50 |
-0.38 |
-0.20 |
-0.18 |
0.07 |
0.22 |
0.41 |
1.16 |
1.46 |
(21.32) |
|
|
t + 10Y |
-0.06 |
-0.17 |
-0.41 |
-0.42 |
-0.13 |
-0.19 |
0.00 |
0.11 |
0.14 |
0.48 |
0.54 |
(8.24) |
|
|
Finland |
t |
-1.61 |
-1.03 |
-0.71 |
-0.43 |
-0.16 |
0.09 |
0.35 |
0.65 |
1.04 |
1.83 |
3.44 |
(231.98) |
|
t + 3Y |
-0.69 |
-0.46 |
-0.34 |
-0.31 |
-0.17 |
-0.06 |
0.19 |
0.41 |
0.54 |
0.98 |
1.67 |
(27.84) |
|
|
t + 10Y |
-0.43 |
-0.10 |
-0.22 |
-0.21 |
-0.07 |
0.02 |
-0.02 |
0.36 |
0.70 |
0.73 |
1.17 |
(12.31) |
|
|
Germany |
t |
-1.72 |
-1.06 |
-0.70 |
-0.41 |
-0.14 |
0.12 |
0.40 |
0.69 |
1.06 |
1.76 |
3.48 |
(591.68) |
|
t + 3Y |
-0.74 |
-0.46 |
-0.33 |
-0.22 |
-0.08 |
0.00 |
0.17 |
0.33 |
0.54 |
0.94 |
1.68 |
(55.67) |
|
|
t + 10Y |
-0.40 |
-0.34 |
-0.20 |
-0.23 |
-0.07 |
0.08 |
0.13 |
0.28 |
0.46 |
0.68 |
1.08 |
(21.31) |
|
|
Great Britain |
t |
-1.73 |
-1.07 |
-0.68 |
-0.38 |
-0.12 |
0.12 |
0.38 |
0.66 |
1.04 |
1.78 |
3.51 |
(678.87) |
|
t + 3Y |
-0.85 |
-0.47 |
-0.26 |
-0.15 |
-0.10 |
0.05 |
0.16 |
0.35 |
0.56 |
1.03 |
1.88 |
(79.38) |
|
|
t + 10Y |
-0.68 |
-0.25 |
-0.13 |
-0.07 |
-0.07 |
0.05 |
0.12 |
0.18 |
0.35 |
0.59 |
1.27 |
(30.70) |
|
|
Greece |
t |
-1.63 |
-1.06 |
-0.73 |
-0.43 |
-0.15 |
0.11 |
0.37 |
0.66 |
1.04 |
1.83 |
3.46 |
(350.98) |
|
t + 3Y |
-0.81 |
-0.60 |
-0.46 |
-0.31 |
-0.23 |
-0.04 |
0.13 |
0.40 |
0.47 |
1.09 |
1.90 |
(34.84) |
|
|
t + 10Y |
-0.49 |
-0.50 |
-0.50 |
-0.24 |
-0.05 |
-0.04 |
0.02 |
0.26 |
0.28 |
0.67 |
1.16 |
(18.73) |
|
|
Hong Kong |
t |
-1.64 |
-1.06 |
-0.72 |
-0.44 |
-0.18 |
0.08 |
0.37 |
0.68 |
1.10 |
1.82 |
3.46 |
(537.86) |
|
t + 3Y |
-0.68 |
-0.57 |
-0.43 |
-0.35 |
-0.27 |
-0.18 |
-0.09 |
0.08 |
0.27 |
0.53 |
1.21 |
(56.55) |
|
|
t + 10Y |
-0.38 |
-0.34 |
-0.33 |
-0.20 |
-0.25 |
-0.34 |
-0.20 |
-0.15 |
0.04 |
0.08 |
0.46 |
(19.61) |
|
|
Hungary |
t |
-1.99 |
-1.73 |
-1.44 |
-1.28 |
-1.17 |
-1.04 |
-0.94 |
-0.85 |
-0.77 |
-0.68 |
1.32 |
(20.61) |
|
t + 3Y |
0.24 |
-0.42 |
-0.34 |
-0.22 |
0.02 |
0.15 |
-1.14 |
-0.12 |
-0.13 |
-0.17 |
-0.93 |
(-4.75) |
|
|
Indonesia |
t |
-1.77 |
-1.07 |
-0.66 |
-0.38 |
-0.12 |
0.15 |
0.41 |
0.69 |
1.04 |
1.73 |
3.49 |
(666.84) |
|
t + 3Y |
-0.81 |
-0.68 |
-0.46 |
-0.34 |
-0.27 |
-0.10 |
0.01 |
0.14 |
0.38 |
1.05 |
1.86 |
(57.31) |
|
|
t + 10Y |
-0.42 |
-0.39 |
-0.18 |
-0.30 |
-0.22 |
-0.03 |
-0.06 |
-0.04 |
0.12 |
0.59 |
1.01 |
(11.19) |
|
|
Ireland |
t |
-1.71 |
-0.98 |
-0.64 |
-0.37 |
-0.13 |
0.11 |
0.39 |
0.66 |
1.03 |
1.70 |
3.40 |
(117.26) |
|
t + 3Y |
-0.27 |
-0.22 |
-0.31 |
0.02 |
0.23 |
-0.02 |
-0.11 |
-0.05 |
0.07 |
0.33 |
0.60 |
(4.03) |
|
|
t + 10Y |
-0.82 |
-0.38 |
-0.24 |
-0.09 |
0.20 |
-0.13 |
-0.07 |
0.24 |
-0.09 |
0.27 |
1.46 |
(8.53) |
|
|
Japan |
t |
-1.64 |
-1.05 |
-0.71 |
-0.43 |
-0.17 |
0.08 |
0.35 |
0.66 |
1.06 |
1.85 |
3.49 |
(716.21) |
|
t + 3Y |
-0.71 |
-0.58 |
-0.45 |
-0.34 |
-0.22 |
-0.09 |
0.06 |
0.24 |
0.49 |
0.96 |
1.67 |
(79.45) |
|
|
t + 10Y |
-0.58 |
-0.48 |
-0.44 |
-0.35 |
-0.28 |
-0.14 |
0.01 |
0.10 |
0.33 |
0.73 |
1.31 |
(56.19) |
|
|
Malaysia |
t |
-1.66 |
-1.02 |
-0.66 |
-0.41 |
-0.18 |
0.07 |
0.32 |
0.65 |
1.09 |
1.83 |
3.50 |
(442.96) |
|
t + 3Y |
-0.66 |
-0.41 |
-0.33 |
-0.28 |
-0.20 |
-0.05 |
0.11 |
0.25 |
0.55 |
1.05 |
1.71 |
(38.23) |
|
|
t + 10Y |
-0.15 |
-0.05 |
-0.11 |
0.02 |
-0.07 |
-0.05 |
0.21 |
0.20 |
0.12 |
0.56 |
0.72 |
(16.92) |
|
|
Malaysia |
t |
-1.71 |
-1.08 |
-0.71 |
-0.41 |
-0.14 |
0.12 |
0.40 |
0.70 |
1.09 |
1.74 |
3.45 |
(582.26) |
|
t + 3Y |
-0.63 |
-0.58 |
-0.48 |
-0.35 |
-0.17 |
-0.10 |
0.11 |
0.25 |
0.49 |
0.92 |
1.55 |
(31.71) |
|
|
t + 10Y |
-0.26 |
-0.29 |
-0.38 |
-0.23 |
-0.22 |
-0.12 |
0.05 |
0.14 |
0.24 |
0.52 |
0.78 |
(40.70) |
|
|
Mexico |
t |
-1.59 |
-1.07 |
-0.72 |
-0.45 |
-0.20 |
0.06 |
0.40 |
0.74 |
1.10 |
1.75 |
3.34 |
(217.21) |
|
t + 3Y |
-0.72 |
-0.75 |
-0.62 |
-0.33 |
-0.25 |
-0.16 |
0.23 |
0.34 |
0.60 |
1.18 |
1.90 |
(29.80) |
|
|
t + 10Y |
-0.10 |
-0.33 |
-0.64 |
-0.47 |
-0.23 |
-0.28 |
-0.11 |
0.12 |
0.28 |
0.58 |
0.67 |
(4.85) |
|
|
Netherlands |
t |
-1.70 |
-1.01 |
-0.65 |
-0.39 |
-0.16 |
0.06 |
0.33 |
0.69 |
1.10 |
1.76 |
3.46 |
(293.37) |
|
t + 3Y |
-0.55 |
-0.37 |
-0.24 |
-0.11 |
-0.11 |
-0.04 |
0.06 |
0.23 |
0.44 |
0.78 |
1.33 |
(12.56) |
|
|
t + 10Y |
-0.54 |
0.06 |
0.27 |
0.04 |
-0.05 |
-0.05 |
0.27 |
0.21 |
0.38 |
0.11 |
0.62 |
(5.76) |
|
|
New Zealand |
t |
-1.67 |
-1.04 |
-0.66 |
-0.37 |
-0.14 |
0.09 |
0.33 |
0.62 |
1.02 |
1.83 |
3.50 |
(277.97) |
|
t + 3Y |
-0.62 |
-0.38 |
-0.37 |
-0.35 |
-0.08 |
0.06 |
0.04 |
0.22 |
0.40 |
0.85 |
1.46 |
(17.71) |
|
|
t + 10Y |
-0.09 |
-0.17 |
-0.45 |
-0.18 |
-0.20 |
0.05 |
0.10 |
0.02 |
0.14 |
0.58 |
0.66 |
(3.97) |
|
|
Norway |
t |
-1.72 |
-1.03 |
-0.65 |
-0.37 |
-0.13 |
0.09 |
0.34 |
0.65 |
1.04 |
1.79 |
3.51 |
(407.33) |
|
t + 3Y |
-0.32 |
-0.37 |
-0.27 |
-0.15 |
-0.14 |
-0.07 |
0.02 |
0.22 |
0.38 |
0.69 |
1.01 |
(17.56) |
|
|
t + 10Y |
-0.39 |
0.00 |
-0.15 |
-0.29 |
-0.08 |
-0.05 |
-0.21 |
-0.10 |
0.26 |
0.55 |
0.94 |
(4.83) |
|
|
Pakistan |
t |
-1.71 |
-1.05 |
-0.69 |
-0.40 |
-0.13 |
0.11 |
0.36 |
0.67 |
1.09 |
1.76 |
3.47 |
(207.29) |
|
t + 3Y |
-0.77 |
-0.60 |
-0.36 |
-0.34 |
-0.21 |
-0.04 |
0.02 |
0.25 |
0.49 |
0.97 |
1.74 |
(19.71) |
|
|
t + 10Y |
-0.59 |
-0.54 |
-0.20 |
-0.21 |
-0.14 |
0.12 |
-0.04 |
0.18 |
0.57 |
0.65 |
1.23 |
(30.93) |
|
|
Peru |
t |
-1.57 |
-1.00 |
-0.68 |
-0.44 |
-0.19 |
0.04 |
0.27 |
0.57 |
1.08 |
1.92 |
3.49 |
(211.13) |
|
t + 3Y |
-0.57 |
-0.64 |
-0.26 |
-0.27 |
-0.14 |
-0.01 |
0.10 |
0.23 |
0.59 |
1.18 |
1.76 |
(33.59) |
|
|
t + 10Y |
-0.38 |
-0.38 |
-0.40 |
-0.35 |
-0.27 |
-0.15 |
-0.06 |
-0.01 |
0.43 |
0.63 |
1.00 |
(5.06) |
|
|
Philippines |
t |
-1.59 |
-1.07 |
-0.71 |
-0.42 |
-0.16 |
0.08 |
0.34 |
0.62 |
1.03 |
1.88 |
3.48 |
(220.86) |
|
t + 3Y |
-0.84 |
-0.68 |
-0.37 |
-0.23 |
-0.14 |
0.05 |
0.24 |
0.28 |
0.52 |
1.06 |
1.90 |
(24.23) |
|
|
t + 10Y |
-0.56 |
-0.45 |
-0.09 |
-0.18 |
-0.01 |
-0.04 |
0.00 |
0.37 |
0.42 |
0.42 |
0.98 |
(13.67) |
|
|
Poland |
t |
-1.71 |
-1.00 |
-0.65 |
-0.38 |
-0.15 |
0.07 |
0.33 |
0.65 |
1.05 |
1.81 |
3.52 |
(260.77) |
|
t + 3Y |
-0.39 |
-0.33 |
-0.32 |
-0.25 |
-0.18 |
-0.14 |
-0.05 |
0.05 |
0.35 |
0.84 |
1.23 |
(32.13) |
|
|
t + 10Y |
0.05 |
-0.03 |
-0.07 |
-0.13 |
0.08 |
0.09 |
0.06 |
0.23 |
0.24 |
0.52 |
0.48 |
(3.85) |
|
|
Portugal |
t |
-1.66 |
-0.97 |
-0.63 |
-0.37 |
-0.14 |
0.07 |
0.31 |
0.61 |
1.01 |
1.80 |
3.46 |
(164.00) |
|
t + 3Y |
-0.63 |
-0.35 |
-0.37 |
-0.24 |
0.02 |
0.04 |
0.01 |
0.25 |
0.33 |
0.80 |
1.43 |
(12.26) |
|
|
t + 10Y |
-0.32 |
-0.20 |
-0.48 |
-0.02 |
0.22 |
0.26 |
-0.30 |
0.05 |
0.17 |
0.31 |
0.63 |
(6.24) |
|
|
Russia |
t |
-1.76 |
-1.01 |
-0.67 |
-0.38 |
-0.11 |
0.13 |
0.39 |
0.68 |
1.04 |
1.71 |
3.47 |
(243.41) |
|
t + 3Y |
-0.76 |
-0.69 |
-0.49 |
-0.42 |
-0.17 |
-0.17 |
-0.04 |
0.25 |
0.28 |
0.66 |
1.41 |
(19.32) |
|
|
t + 10Y |
-0.15 |
-0.26 |
0.00 |
-0.20 |
-0.23 |
0.09 |
-0.15 |
0.81 |
0.32 |
0.45 |
0.59 |
(3.66) |
|
|
Singapore |
t |
-1.72 |
-1.06 |
-0.69 |
-0.40 |
-0.15 |
0.11 |
0.39 |
0.70 |
1.08 |
1.75 |
3.47 |
(488.51) |
|
t + 3Y |
-0.47 |
-0.43 |
-0.42 |
-0.32 |
-0.19 |
-0.06 |
0.00 |
0.12 |
0.34 |
0.67 |
1.14 |
(27.17) |
|
|
t + 10Y |
-0.29 |
-0.20 |
-0.17 |
-0.08 |
-0.10 |
-0.01 |
0.04 |
-0.01 |
0.22 |
0.24 |
0.54 |
(8.69) |
|
|
South Africa |
t |
-1.71 |
-1.06 |
-0.67 |
-0.38 |
-0.12 |
0.11 |
0.36 |
0.65 |
1.03 |
1.80 |
3.51 |
(422.57) |
|
t + 3Y |
-0.68 |
-0.57 |
-0.39 |
-0.19 |
-0.10 |
-0.02 |
0.15 |
0.24 |
0.42 |
0.82 |
1.50 |
(39.46) |
|
|
t + 10Y |
-0.38 |
-0.24 |
-0.25 |
-0.27 |
-0.18 |
-0.07 |
0.07 |
0.23 |
0.28 |
0.27 |
0.62 |
(8.49) |
|
|
South Korea |
t |
-1.67 |
-1.08 |
-0.73 |
-0.44 |
-0.17 |
0.10 |
0.39 |
0.71 |
1.11 |
1.77 |
3.44 |
(715.74) |
|
t + 3Y |
-0.79 |
-0.69 |
-0.59 |
-0.42 |
-0.31 |
-0.20 |
-0.08 |
0.14 |
0.31 |
0.73 |
1.52 |
(26.20) |
|
|
t + 10Y |
-0.52 |
-0.47 |
-0.32 |
-0.33 |
-0.36 |
-0.22 |
-0.19 |
-0.01 |
0.01 |
0.33 |
0.86 |
(47.10) |
|
|
Spain |
t |
-1.67 |
-1.04 |
-0.69 |
-0.40 |
-0.12 |
0.12 |
0.35 |
0.64 |
1.03 |
1.81 |
3.48 |
(269.31) |
|
t + 3Y |
-0.65 |
-0.51 |
-0.38 |
-0.23 |
-0.13 |
-0.03 |
0.12 |
0.34 |
0.60 |
1.02 |
1.66 |
(21.39) |
|
|
t + 10Y |
-0.49 |
-0.14 |
0.13 |
0.07 |
-0.12 |
0.08 |
0.09 |
0.46 |
0.25 |
0.67 |
1.16 |
(12.20) |
|
|
Spain |
t |
-1.67 |
-1.04 |
-0.69 |
-0.40 |
-0.12 |
0.12 |
0.35 |
0.64 |
1.03 |
1.81 |
3.48 |
(269.31) |
|
t + 3Y |
-0.65 |
-0.51 |
-0.38 |
-0.23 |
-0.13 |
-0.03 |
0.12 |
0.34 |
0.60 |
1.02 |
1.66 |
(21.39) |
|
|
t + 10Y |
-0.49 |
-0.14 |
0.13 |
0.07 |
-0.12 |
0.08 |
0.09 |
0.46 |
0.25 |
0.67 |
1.16 |
(12.20) |
|
|
Sweden |
t |
-1.68 |
-1.06 |
-0.71 |
-0.40 |
-0.13 |
0.12 |
0.37 |
0.66 |
1.03 |
1.81 |
3.49 |
(459.65) |
|
t + 3Y |
-0.41 |
-0.32 |
-0.30 |
-0.22 |
-0.05 |
0.06 |
0.24 |
0.39 |
0.59 |
0.99 |
1.40 |
(26.67) |
|
|
t + 10Y |
0.04 |
-0.13 |
-0.09 |
0.11 |
0.31 |
0.30 |
0.37 |
0.44 |
0.39 |
0.76 |
0.72 |
(11.67) |
|
|
Switzerland |
t |
-1.56 |
-1.03 |
-0.74 |
-0.47 |
-0.21 |
0.06 |
0.34 |
0.67 |
1.09 |
1.87 |
3.43 |
(226.51) |
|
t + 3Y |
-0.76 |
-0.52 |
-0.45 |
-0.20 |
-0.14 |
-0.03 |
0.15 |
0.36 |
0.64 |
1.14 |
1.90 |
(30.43) |
|
|
t + 10Y |
-0.66 |
-0.51 |
-0.32 |
-0.17 |
-0.09 |
0.03 |
0.13 |
0.34 |
0.34 |
0.74 |
1.40 |
(20.48) |
|
|
Switzerland |
t |
-1.56 |
-1.03 |
-0.74 |
-0.47 |
-0.21 |
0.06 |
0.34 |
0.67 |
1.09 |
1.87 |
3.43 |
(226.51) |
|
t + 3Y |
-0.76 |
-0.52 |
-0.45 |
-0.20 |
-0.14 |
-0.03 |
0.15 |
0.36 |
0.64 |
1.14 |
1.90 |
(30.43) |
|
|
t + 10Y |
-0.66 |
-0.51 |
-0.32 |
-0.17 |
-0.09 |
0.03 |
0.13 |
0.34 |
0.34 |
0.74 |
1.40 |
(20.48) |
|
|
Taiwan |
t |
-1.64 |
-1.08 |
-0.73 |
-0.45 |
-0.17 |
0.10 |
0.39 |
0.71 |
1.09 |
1.79 |
3.43 |
(741.48) |
|
t + 3Y |
-0.63 |
-0.58 |
-0.48 |
-0.37 |
-0.28 |
-0.07 |
0.08 |
0.27 |
0.50 |
0.93 |
1.56 |
(59.58) |
|
|
t + 10Y |
-0.32 |
-0.33 |
-0.28 |
-0.35 |
-0.26 |
-0.05 |
-0.03 |
0.03 |
0.20 |
0.40 |
0.72 |
(17.31) |
|
|
Thailand |
t |
-1.74 |
-1.06 |
-0.67 |
-0.38 |
-0.14 |
0.12 |
0.39 |
0.69 |
1.05 |
1.76 |
3.50 |
(557.28) |
|
t + 3Y |
-0.74 |
-0.61 |
-0.45 |
-0.27 |
-0.22 |
-0.02 |
0.06 |
0.20 |
0.46 |
0.92 |
1.66 |
(33.17) |
|
|
t + 10Y |
-0.35 |
-0.23 |
-0.21 |
-0.22 |
-0.20 |
-0.20 |
-0.05 |
0.09 |
0.05 |
0.19 |
0.53 |
(6.10) |
|
|
Turkey |
t |
-1.71 |
-1.06 |
-0.69 |
-0.42 |
-0.12 |
0.15 |
0.40 |
0.67 |
1.05 |
1.75 |
3.47 |
(242.05) |
|
t + 3Y |
-0.81 |
-0.43 |
-0.41 |
-0.29 |
-0.07 |
0.09 |
0.18 |
0.38 |
0.57 |
0.92 |
1.73 |
(46.10) |
|
|
t + 10Y |
-0.29 |
-0.12 |
-0.21 |
0.02 |
0.00 |
0.11 |
0.16 |
0.29 |
0.27 |
0.41 |
0.70 |
(11.88) |
|
|
United States |
t |
-1.60 |
-1.01 |
-0.73 |
-0.47 |
-0.21 |
0.06 |
0.34 |
0.67 |
1.08 |
1.87 |
3.47 |
(313.63) |
|
t + 3Y |
-0.76 |
-0.62 |
-0.49 |
-0.36 |
-0.20 |
-0.07 |
0.10 |
0.30 |
0.56 |
1.01 |
1.77 |
(56.42) |
|
|
t + 10Y |
-0.42 |
-0.46 |
-0.43 |
-0.27 |
-0.10 |
-0.05 |
0.00 |
0.17 |
0.30 |
0.58 |
1.00 |
(58.65) |
Table A-2: Regressions of quality-sorted
portfolio excess returns on risk factors
|
Country |
|
P1 |
P2 |
P3 |
P4 |
P5 |
P6 |
P7 |
P8 |
P9 |
P10 |
H-L |
|
United States |
Excess Return |
-1.86 |
-1.13 |
-0.88 |
-0.90 |
0.33 |
-0.60 |
-0.22 |
-0.39 |
-0.20 |
-0.11 |
1.75 |
|
|
(-1.73) |
(-1.14) |
(-0.97) |
(-0.99) |
(0.26) |
(-0.74) |
(-0.26) |
(-0.50) |
(-0.26) |
(-0.14) |
(3.68) |
|
|
CAPM α |
-1.18 |
-0.76 |
-0.29 |
-0.32 |
0.03 |
-0.07 |
0.19 |
0.06 |
-0.10 |
0.38 |
1.55 |
|
|
|
(-2.36) |
(-1.31) |
(-0.69) |
(-0.74) |
(0.04) |
(-0.20) |
(0.54) |
(0.16) |
(-0.36) |
(1.45) |
(3.34) |
|
|
3-factor α |
-1.06 |
-0.25 |
-0.48 |
-0.26 |
-0.48 |
-0.07 |
-0.01 |
0.02 |
-0.19 |
0.08 |
1.14 |
|
|
|
(-2.55) |
(-0.49) |
(-1.13) |
(-0.67) |
(-0.97) |
(-0.24) |
(-0.03) |
(0.04) |
(-0.58) |
(0.27) |
(3.60) |
|
|
4-factor α |
-0.95 |
-0.68 |
-0.52 |
0.00 |
-0.41 |
-0.07 |
-0.15 |
-0.12 |
-0.31 |
-0.18 |
0.77 |
|
|
|
(-2.57) |
(-1.08) |
(-1.37) |
(-0.02) |
(-0.94) |
(-0.22) |
(-0.42) |
(-0.39) |
(-0.91) |
(-0.67) |
(3.23) |
|
|
Sharpe Ratio |
-0.63 |
-0.43 |
-0.36 |
-0.37 |
0.11 |
-0.27 |
-0.10 |
-0.19 |
-0.10 |
-0.05 |
1.96 |
|
|
Information Ratio |
(-1.14) |
(-0.75) |
(-0.77) |
(-0.01) |
(-0.55) |
(-0.10) |
(-0.21) |
(-0.16) |
(-0.42) |
(-0.30) |
(1.04) |
|
|
Great Britain |
Excess Return |
-0.77 |
-0.41 |
-0.22 |
-0.03 |
0.05 |
0.15 |
0.42 |
0.58 |
0.46 |
0.81 |
1.58 |
|
|
(-1.64) |
(-0.98) |
(-0.50) |
(-0.06) |
(0.11) |
(0.34) |
(0.99) |
(1.34) |
(1.21) |
(2.21) |
(8.59) |
|
|
CAPM α |
-1.11 |
-0.56 |
-0.58 |
-0.53 |
-0.21 |
-0.31 |
0.42 |
0.34 |
0.23 |
1.12 |
2.23 |
|
|
|
(-3.77) |
(-2.22) |
(-1.99) |
(-1.68) |
(-0.80) |
(-1.14) |
(1.78) |
(1.49) |
(0.97) |
(4.18) |
(8.97) |
|
|
3-factor α |
-1.04 |
-0.37 |
-0.42 |
-0.47 |
-0.10 |
-0.24 |
0.47 |
0.34 |
0.35 |
1.19 |
2.23 |
|
|
|
(-4.11) |
(-1.63) |
(-1.60) |
(-1.65) |
(-0.44) |
(-0.81) |
(1.73) |
(1.33) |
(1.43) |
(4.57) |
(9.18) |
|
|
4-factor α |
-1.05 |
-0.52 |
-0.49 |
-0.55 |
-0.18 |
-0.33 |
0.47 |
0.38 |
0.33 |
1.17 |
2.22 |
|
|
|
(-4.07) |
(-2.25) |
(-1.91) |
(-1.93) |
(-0.72) |
(-1.10) |
(1.81) |
(1.70) |
(1.40) |
(4.50) |
(9.68) |
|
|
Sharpe Ratio |
-0.55 |
-0.36 |
-0.18 |
-0.02 |
0.04 |
0.13 |
0.37 |
0.48 |
0.44 |
0.79 |
2.32 |
|
|
Information Ratio |
(-1.25) |
(-0.66) |
(-0.66) |
(-0.69) |
(-0.23) |
(-0.42) |
(0.63) |
(0.53) |
(0.47) |
(1.67) |
(2.90) |
|
|
Japan |
Excess Return |
0.58 |
0.68 |
0.65 |
0.71 |
0.61 |
0.70 |
0.64 |
0.80 |
0.70 |
0.74 |
0.16 |
|
|
(1.15) |
(1.49) |
(1.43) |
(0.00) |
(1.26) |
(1.46) |
(1.33) |
(0.00) |
(0.00) |
(1.38) |
(1.24) |
|
|
CAPM α |
0.14 |
0.32 |
0.47 |
0.40 |
0.37 |
0.62 |
0.45 |
0.55 |
0.48 |
0.73 |
0.59 |
|
|
|
(0.51) |
(1.29) |
(1.81) |
(1.55) |
(1.53) |
(2.37) |
(1.96) |
(2.50) |
(2.21) |
(2.84) |
(0.00) |
|
|
3-factor α |
0.20 |
0.35 |
0.52 |
0.46 |
0.42 |
0.64 |
0.51 |
0.59 |
0.61 |
0.71 |
0.51 |
|
|
|
(0.79) |
(1.47) |
(2.10) |
(1.88) |
(1.75) |
(2.54) |
(2.26) |
(2.82) |
(2.95) |
(3.05) |
(2.96) |
|
|
4-factor α |
-0.06 |
0.30 |
0.37 |
0.35 |
0.27 |
0.43 |
0.39 |
0.43 |
0.46 |
0.63 |
0.68 |
|
|
(-0.23) |
(1.24) |
(1.75) |
(1.41) |
(1.13) |
(1.70) |
(1.79) |
(1.98) |
(2.04) |
(2.77) |
(3.42) |
||
|
Sharpe Ratio |
0.38 |
0.49 |
0.47 |
0.50 |
0.42 |
0.49 |
0.44 |
0.53 |
0.45 |
0.45 |
0.28 |
|
|
Information Ratio |
(-0.08) |
(0.43) |
(0.61) |
(0.53) |
(0.40) |
(0.65) |
(0.59) |
(0.64) |
(0.68) |
(0.95) |
(1.08) |
|
|
Hong Kong |
Excess Return |
0.96 |
0.91 |
0.94 |
0.79 |
1.00 |
0.97 |
0.90 |
0.81 |
0.93 |
0.87 |
-0.10 |
|
(1.22) |
(1.15) |
(1.20) |
(1.00) |
(1.29) |
(1.27) |
(1.23) |
(1.10) |
(1.37) |
(1.26) |
(-0.38) |
||
|
CAPM α |
1.23 |
0.92 |
0.72 |
-0.03 |
0.97 |
1.80 |
0.87 |
0.93 |
0.74 |
1.15 |
-0.08 |
|
|
(1.68) |
(1.88) |
(1.45) |
(-0.04) |
(1.63) |
(1.59) |
(2.08) |
(2.25) |
(1.70) |
(2.54) |
(-0.14) |
||
|
3-factor α |
0.62 |
0.68 |
0.77 |
0.29 |
0.71 |
0.86 |
0.69 |
0.79 |
0.44 |
0.64 |
0.02 |
|
|
(1.85) |
(1.68) |
(2.08) |
(0.58) |
(1.83) |
(1.93) |
(1.81) |
(2.25) |
(1.21) |
(1.39) |
(0.06) |
||
|
4-factor α |
0.60 |
0.46 |
0.76 |
0.66 |
0.69 |
0.66 |
0.72 |
0.68 |
0.46 |
0.49 |
-0.11 |
|
|
(1.74) |
(1.05) |
(1.95) |
(1.50) |
(1.88) |
(1.58) |
(1.84) |
(1.92) |
(1.26) |
(1.09) |
(-0.25) |
||
|
Sharpe Ratio |
0.46 |
0.43 |
0.47 |
0.38 |
0.49 |
0.47 |
0.49 |
0.43 |
0.52 |
0.46 |
-0.11 |
|
|
Information Ratio |
(0.45) |
(0.35) |
(0.60) |
(0.50) |
(0.57) |
(0.57) |
(0.67) |
(0.67) |
(0.43) |
(0.42) |
(-0.06) |
|
|
China |
Excess Return |
1.54 |
1.54 |
1.70 |
1.74 |
1.70 |
1.72 |
1.68 |
1.72 |
1.80 |
1.90 |
0.36 |
|
(1.58) |
(1.68) |
(1.82) |
(1.89) |
(1.79) |
(1.85) |
(1.77) |
(1.82) |
(1.98) |
(2.21) |
(1.43) |
||
|
CAPM α |
0.10 |
-0.09 |
0.08 |
0.44 |
0.39 |
0.78 |
0.63 |
1.46 |
0.97 |
1.29 |
1.20 |
|
|
(0.22) |
(-0.20) |
(0.19) |
(0.92) |
(0.93) |
(1.54) |
(1.45) |
(2.83) |
(2.77) |
(2.52) |
(2.11) |
||
|
3-factor α |
0.36 |
-0.10 |
0.40 |
0.58 |
0.97 |
0.26 |
0.09 |
1.27 |
1.21 |
0.79 |
0.43 |
|
|
(0.76) |
(-0.25) |
(1.01) |
(1.42) |
(2.32) |
(0.55) |
(0.15) |
(2.91) |
(2.69) |
(1.92) |
(0.79) |
||
|
4-factor α |
0.70 |
-1.81 |
0.53 |
0.63 |
1.04 |
0.04 |
-0.93 |
1.11 |
2.26 |
1.34 |
0.63 |
|
|
(1.12) |
(-1.01) |
(1.33) |
(1.51) |
(2.47) |
(0.05) |
(-0.70) |
(2.54) |
(1.70) |
(2.05) |
(1.31) |
||
|
Sharpe Ratio |
0.54 |
0.55 |
0.62 |
0.64 |
0.62 |
0.64 |
0.61 |
0.64 |
0.68 |
0.78 |
0.32 |
|
|
Information Ratio |
(0.28) |
(-0.25) |
(0.32) |
(0.40) |
(0.56) |
(0.01) |
(-0.18) |
(0.66) |
(0.52) |
(0.62) |
(0.31) |
|
|
Canada |
Excess Return |
2.04 |
1.79 |
1.50 |
1.39 |
1.37 |
1.08 |
1.20 |
1.30 |
1.15 |
1.35 |
-0.69 |
|
(2.61) |
(2.27) |
(1.74) |
(1.80) |
(1.86) |
(1.55) |
(1.77) |
(2.01) |
(1.98) |
(2.68) |
(-1.64) |
||
|
CAPM α |
1.70 |
1.69 |
1.13 |
1.32 |
0.95 |
1.16 |
1.02 |
1.14 |
1.05 |
1.39 |
-0.32 |
|
|
(3.00) |
(2.87) |
(2.23) |
(2.80) |
(2.20) |
(3.20) |
(3.39) |
(3.94) |
(4.00) |
(5.30) |
(-0.66) |
||
|
3-factor α |
0.89 |
1.06 |
0.55 |
1.04 |
0.68 |
0.70 |
0.76 |
0.82 |
0.89 |
1.14 |
0.26 |
|
|
(2.07) |
(2.97) |
(1.42) |
(3.25) |
(2.06) |
(2.29) |
(2.46) |
(2.59) |
(3.31) |
(5.06) |
(0.67) |
||
|
4-factor α |
0.84 |
0.93 |
0.47 |
0.85 |
0.77 |
0.75 |
0.84 |
0.99 |
0.88 |
1.14 |
0.30 |
|
|
(1.95) |
(2.55) |
(1.26) |
(2.58) |
(2.42) |
(2.47) |
(2.97) |
(3.43) |
(3.38) |
(4.93) |
(0.80) |
||
|
Sharpe Ratio |
1.05 |
0.90 |
0.70 |
0.67 |
0.68 |
0.57 |
0.62 |
0.73 |
0.70 |
0.92 |
-0.58 |
|
|
Information Ratio |
(0.67) |
(0.75) |
(0.40) |
(0.83) |
(0.69) |
(0.81) |
(0.98) |
(1.17) |
(1.09) |
(1.52) |
(0.23) |
|
|
Germany |
Excess Return |
-0.11 |
0.22 |
0.43 |
0.80 |
0.30 |
0.62 |
0.75 |
0.79 |
0.91 |
0.96 |
1.07 |
|
(-0.23) |
(0.57) |
(1.12) |
(2.15) |
(0.86) |
(1.64) |
(2.01) |
(2.22) |
(2.30) |
(3.06) |
(3.73) |
||
|
CAPM α |
-0.50 |
0.58 |
0.48 |
0.82 |
0.11 |
0.88 |
0.90 |
0.05 |
0.82 |
0.72 |
1.21 |
|
|
(-0.91) |
(1.86) |
(1.38) |
(2.95) |
(0.40) |
(2.10) |
(4.13) |
(0.08) |
(2.81) |
(3.49) |
(2.22) |
||
|
3-factor α |
0.06 |
0.44 |
0.51 |
0.52 |
0.54 |
0.70 |
1.11 |
0.57 |
0.85 |
1.09 |
1.02 |
|
|
(0.18) |
(1.79) |
(1.88) |
(2.31) |
(2.10) |
(2.81) |
(5.82) |
(3.00) |
(3.27) |
(6.05) |
(2.71) |
||
|
4-factor α |
0.22 |
0.31 |
0.63 |
0.54 |
0.50 |
0.60 |
0.99 |
0.54 |
0.87 |
1.09 |
0.87 |
|
|
(0.65) |
(1.18) |
(2.42) |
(2.42) |
(1.92) |
(2.30) |
(5.48) |
(3.05) |
(3.31) |
(6.14) |
(2.49) |
||
|
Sharpe Ratio |
-0.07 |
0.19 |
0.39 |
0.77 |
0.29 |
0.60 |
0.74 |
0.84 |
0.86 |
0.98 |
0.98 |
|
|
Information Ratio |
(0.19) |
(0.34) |
(0.73) |
(0.66) |
(0.61) |
(0.74) |
(1.34) |
(0.71) |
(1.04) |
(1.36) |
(0.65) |
|
|
Australia |
Excess Return |
0.36 |
0.82 |
0.60 |
0.83 |
0.62 |
0.95 |
0.67 |
0.84 |
1.01 |
0.81 |
0.45 |
|
(0.67) |
(1.29) |
(0.90) |
(0.90) |
(0.91) |
(1.59) |
(1.20) |
(1.53) |
(1.93) |
(1.72) |
(1.93) |
||
|
CAPM α |
-0.36 |
0.40 |
-0.16 |
0.10 |
0.52 |
0.90 |
0.72 |
0.74 |
0.87 |
0.62 |
0.99 |
|
|
(-1.09) |
(1.09) |
(-0.42) |
(0.22) |
(1.33) |
(2.47) |
(2.47) |
(2.21) |
(2.64) |
(1.98) |
(2.87) |
||
|
3-factor α |
-0.63 |
0.41 |
-0.49 |
-0.09 |
0.35 |
0.69 |
0.59 |
0.78 |
0.91 |
0.72 |
1.34 |
|
|
(-1.94) |
(1.15) |
(-1.41) |
(-0.22) |
(1.01) |
(2.27) |
(2.05) |
(2.47) |
(2.73) |
(2.34) |
(4.17) |
||
|
4-factor α |
-0.68 |
0.33 |
-0.48 |
-0.07 |
0.38 |
0.75 |
0.53 |
0.74 |
0.90 |
0.65 |
1.34 |
|
|
(-2.11) |
(0.93) |
(-1.41) |
(-0.20) |
(1.11) |
(2.50) |
(1.90) |
(2.32) |
(2.76) |
(2.12) |
(4.11) |
||
|
Sharpe Ratio |
0.23 |
0.48 |
0.34 |
0.47 |
0.35 |
0.56 |
0.44 |
0.57 |
0.75 |
0.60 |
0.57 |
|
|
Information Ratio |
(-0.57) |
(0.26) |
(-0.39) |
(-0.06) |
(0.33) |
(0.72) |
(0.60) |
(0.85) |
(1.04) |
(0.76) |
(1.08) |
|
|
South Korea |
Excess Return |
0.47 |
0.93 |
1.24 |
1.32 |
1.22 |
1.22 |
1.34 |
1.41 |
1.46 |
1.12 |
0.65 |
|
(0.86) |
(1.74) |
(2.37) |
(2.50) |
(2.42) |
(2.44) |
(2.64) |
(2.75) |
(2.93) |
(2.34) |
(2.18) |
||
|
CAPM α |
-0.09 |
-0.07 |
0.19 |
0.91 |
0.75 |
0.90 |
0.25 |
0.88 |
0.65 |
1.49 |
1.58 |
|
|
(-0.13) |
(-0.12) |
(0.35) |
(2.25) |
(1.52) |
(1.87) |
(0.60) |
(2.31) |
(1.88) |
(3.44) |
(2.50) |
||
|
3-factor α |
-0.07 |
0.07 |
0.57 |
1.19 |
1.09 |
1.11 |
0.08 |
0.91 |
0.67 |
1.33 |
1.40 |
|
|
(-0.11) |
(0.14) |
(1.29) |
(3.39) |
(2.95) |
(3.03) |
(0.26) |
(2.70) |
(2.00) |
(3.38) |
(2.26) |
||
|
4-factor α |
-0.27 |
0.04 |
0.39 |
1.03 |
1.05 |
1.01 |
-0.15 |
0.56 |
0.52 |
1.17 |
1.44 |
|
|
(-0.51) |
(0.08) |
(0.92) |
(3.70) |
(3.68) |
(3.22) |
(-0.50) |
(1.84) |
(1.55) |
(3.46) |
(2.34) |
||
|
Sharpe Ratio |
0.24 |
0.51 |
0.65 |
0.75 |
0.68 |
0.66 |
0.74 |
0.78 |
0.80 |
0.64 |
0.73 |
|
|
Information Ratio |
(-0.15) |
(0.03) |
(0.26) |
(0.98) |
(0.98) |
(0.93) |
(-0.12) |
(0.52) |
(0.40) |
(0.98) |
(0.77) |
|
|
Switzerland |
Excess Return |
-0.03 |
-0.07 |
0.17 |
0.65 |
0.13 |
0.48 |
0.45 |
0.44 |
0.60 |
0.91 |
0.95 |
|
(-0.09) |
(-0.20) |
(0.50) |
(1.81) |
(0.34) |
(1.41) |
(1.28) |
(1.22) |
(1.53) |
(2.27) |
(3.24) |
||
|
CAPM α |
-0.58 |
-0.20 |
0.01 |
0.20 |
0.22 |
0.20 |
0.10 |
0.33 |
0.40 |
0.51 |
1.09 |
|
|
(-1.72) |
(-0.73) |
(0.06) |
(0.86) |
(0.66) |
(0.87) |
(0.40) |
(1.06) |
(1.49) |
(1.91) |
(2.81) |
||
|
3-factor α |
-0.02 |
0.48 |
0.48 |
0.40 |
0.69 |
0.41 |
0.35 |
0.53 |
0.54 |
0.38 |
0.40 |
|
|
(-0.06) |
(1.87) |
(2.26) |
(1.65) |
(2.46) |
(1.70) |
(1.21) |
(1.56) |
(2.02) |
(1.54) |
(1.13) |
||
|
4-factor α |
-0.23 |
0.38 |
0.31 |
0.28 |
0.62 |
0.62 |
-0.18 |
0.36 |
0.43 |
0.26 |
0.50 |
|
|
(-0.65) |
(1.24) |
(1.45) |
(1.24) |
(2.30) |
(2.34) |
(-0.54) |
(1.17) |
(1.53) |
(1.04) |
(1.21) |
||
|
Sharpe Ratio |
-0.03 |
-0.07 |
0.17 |
0.60 |
0.11 |
0.47 |
0.38 |
0.39 |
0.51 |
0.73 |
1.02 |
|
|
Information Ratio |
(-0.15) |
(0.39) |
(0.39) |
(0.32) |
(0.62) |
(0.56) |
(-0.14) |
(0.30) |
(0.43) |
(0.20) |
(0.27) |
|
|
Russia |
Excess Return |
0.89 |
1.13 |
1.64 |
1.36 |
1.52 |
1.05 |
1.50 |
1.28 |
1.59 |
1.39 |
0.50 |
|
(0.92) |
(1.29) |
(2.04) |
(1.45) |
(2.23) |
(1.69) |
(1.78) |
(1.97) |
(2.41) |
(2.01) |
(0.74) |
||
|
CAPM α |
0.71 |
0.50 |
1.21 |
43.50 |
1.62 |
1.33 |
2.36 |
1.64 |
1.09 |
2.52 |
1.80 |
|
|
(0.62) |
(0.60) |
(1.73) |
(1.14) |
(2.65) |
(2.62) |
(2.17) |
(2.12) |
(1.78) |
(1.47) |
(0.77) |
||
|
3-factor α |
-1.20 |
1.05 |
1.57 |
-1.36 |
30.26 |
1.03 |
1.50 |
3.99 |
0.87 |
0.94 |
2.14 |
|
|
(-1.47) |
(0.77) |
(1.90) |
(-1.12) |
(1.15) |
(1.18) |
(1.88) |
(1.28) |
(1.07) |
(0.51) |
(0.98) |
||
|
4-factor α |
-1.14 |
3.50 |
0.89 |
-0.39 |
1.19 |
-1.21 |
0.74 |
1.10 |
-0.21 |
-4.25 |
-3.11 |
|
|
(-1.07) |
(1.33) |
(1.76) |
(-0.77) |
(1.93) |
(-0.76) |
(1.39) |
(1.43) |
(-0.23) |
(-0.97) |
(-0.85) |
||
|
Sharpe Ratio |
0.37 |
0.55 |
0.87 |
0.63 |
0.83 |
0.61 |
0.76 |
0.74 |
0.95 |
0.79 |
0.25 |
|
|
Information Ratio |
(-0.30) |
(0.36) |
(0.36) |
(-0.20) |
(0.43) |
(-0.19) |
(0.23) |
(0.33) |
(-0.04) |
(-0.25) |
(-0.21) |
|
|
Spain |
Excess Return |
-0.72 |
-0.14 |
0.41 |
0.14 |
0.21 |
0.46 |
0.62 |
0.33 |
0.73 |
1.01 |
1.73 |
|
(-1.32) |
(-0.25) |
(0.87) |
(0.32) |
(0.44) |
(1.07) |
(1.46) |
(0.84) |
(2.10) |
(2.96) |
(4.76) |
||
|
CAPM α |
-1.15 |
-0.32 |
0.66 |
0.12 |
-0.09 |
0.60 |
0.30 |
0.18 |
0.09 |
1.33 |
2.48 |
|
|
(-3.04) |
(-0.78) |
(1.77) |
(0.38) |
(-0.22) |
(1.55) |
(0.89) |
(0.61) |
(0.30) |
(4.72) |
(5.98) |
||
|
3-factor α |
-0.38 |
0.18 |
0.48 |
0.28 |
0.17 |
0.38 |
0.33 |
0.09 |
0.03 |
0.65 |
1.03 |
|
|
(-0.78) |
(0.46) |
(1.56) |
(0.85) |
(0.45) |
(1.24) |
(1.17) |
(0.31) |
(0.08) |
(1.86) |
(1.79) |
||
|
4-factor α |
-0.37 |
0.29 |
0.53 |
0.11 |
0.09 |
0.45 |
0.20 |
0.34 |
0.05 |
0.39 |
0.76 |
|
|
(-0.75) |
(0.79) |
(1.53) |
(0.30) |
(0.23) |
(1.83) |
(0.65) |
(1.06) |
(0.15) |
(1.00) |
(1.35) |
||
|
Sharpe Ratio |
-0.44 |
-0.08 |
0.29 |
0.10 |
0.14 |
0.36 |
0.55 |
0.29 |
0.68 |
0.94 |
1.56 |
|
|
Information Ratio |
(-0.18) |
(0.18) |
(0.41) |
(0.07) |
(0.07) |
(0.33) |
(0.20) |
(0.36) |
(0.04) |
(0.30) |
(0.32) |
|
|
Sweden |
Excess Return |
-0.72 |
0.38 |
0.24 |
0.29 |
0.66 |
0.74 |
0.95 |
1.02 |
1.10 |
1.05 |
1.76 |
|
(-1.42) |
(0.66) |
(0.43) |
(0.57) |
(1.47) |
(1.84) |
(2.17) |
(2.44) |
(2.60) |
(2.58) |
(5.45) |
||
|
CAPM α |
-1.02 |
1.14 |
-0.75 |
-0.47 |
0.68 |
0.12 |
0.59 |
0.84 |
0.74 |
1.09 |
2.12 |
|
|
(-1.78) |
(1.85) |
(-1.35) |
(-0.93) |
(1.54) |
(0.27) |
(1.37) |
(2.01) |
(1.49) |
(2.98) |
(4.52) |
||
|
3-factor α |
-0.84 |
1.02 |
0.27 |
0.13 |
0.58 |
0.75 |
1.04 |
1.22 |
1.37 |
1.60 |
2.44 |
|
|
(-1.53) |
(2.13) |
(0.51) |
(0.24) |
(1.33) |
(1.62) |
(2.00) |
(3.72) |
(3.33) |
(4.37) |
(4.53) |
||
|
4-factor α |
-1.13 |
0.84 |
0.24 |
0.17 |
0.49 |
0.57 |
0.50 |
0.84 |
1.27 |
1.27 |
2.40 |
|
|
(-1.93) |
(1.94) |
(0.45) |
(0.35) |
(1.13) |
(1.22) |
(0.90) |
(2.45) |
(3.27) |
(3.64) |
(4.39) |
||
|
Sharpe Ratio |
-0.41 |
0.25 |
0.14 |
0.21 |
0.47 |
0.56 |
0.72 |
0.80 |
0.89 |
0.83 |
1.44 |
|
|
Information Ratio |
(-0.56) |
(0.54) |
(0.13) |
(0.10) |
(0.31) |
(0.34) |
(0.30) |
(0.61) |
(0.92) |
(0.88) |
(1.23) |
|
|
Malaysia |
Excess Return |
0.01 |
0.42 |
0.62 |
0.83 |
0.69 |
0.67 |
0.95 |
0.84 |
1.01 |
0.84 |
0.83 |
|
(0.03) |
(1.00) |
(1.45) |
(2.10) |
(0.00) |
(1.66) |
(2.39) |
(2.08) |
(2.69) |
(2.31) |
(3.96) |
||
|
CAPM α |
-0.12 |
0.48 |
0.84 |
0.90 |
0.74 |
0.98 |
0.97 |
0.90 |
1.22 |
1.05 |
1.17 |
|
|
(-0.34) |
(1.19) |
(2.91) |
(2.53) |
(2.31) |
(3.14) |
(4.02) |
(3.28) |
(4.51) |
(0.00) |
(2.76) |
||
|
3-factor α |
-0.23 |
0.26 |
0.97 |
0.96 |
0.88 |
1.13 |
0.98 |
1.05 |
1.25 |
0.85 |
1.08 |
|
|
(-0.63) |
(0.89) |
(3.26) |
(3.26) |
(0.00) |
(4.23) |
(4.29) |
(4.19) |
(4.99) |
(0.00) |
(2.49) |
||
|
4-factor α |
-0.29 |
0.25 |
1.04 |
0.83 |
0.77 |
1.06 |
0.75 |
0.92 |
1.09 |
0.69 |
0.98 |
|
|
(-0.82) |
(0.87) |
(3.54) |
(0.00) |
(2.34) |
(4.07) |
(3.19) |
(0.00) |
(4.46) |
(3.08) |
(2.27) |
||
|
Sharpe Ratio |
0.01 |
0.27 |
0.42 |
0.60 |
0.48 |
0.50 |
0.72 |
0.65 |
0.81 |
0.68 |
1.07 |
|
|
Information Ratio |
(-0.24) |
(0.25) |
(1.10) |
(0.87) |
(0.79) |
(1.30) |
(1.03) |
(1.18) |
(1.45) |
(0.98) |
(0.77) |
|
|
Taiwan |
Excess Return |
0.36 |
0.65 |
0.81 |
0.92 |
0.92 |
0.97 |
0.92 |
0.98 |
0.90 |
0.95 |
0.58 |
|
(0.63) |
(1.16) |
(1.49) |
(1.72) |
(1.79) |
(1.74) |
(1.69) |
(1.76) |
(1.63) |
(1.77) |
(2.26) |
||
|
CAPM α |
-0.33 |
-0.37 |
-0.01 |
0.65 |
0.21 |
0.70 |
0.12 |
0.49 |
0.71 |
0.50 |
0.83 |
|
|
(-1.16) |
(-1.45) |
(-0.03) |
(1.81) |
(0.67) |
(1.92) |
(0.53) |
(1.66) |
(2.72) |
(1.61) |
(2.47) |
||
|
3-factor α |
-0.40 |
-0.31 |
-0.01 |
0.55 |
0.15 |
0.81 |
0.08 |
0.48 |
0.69 |
0.58 |
0.98 |
|
|
(-1.74) |
(-1.47) |
(-0.05) |
(1.81) |
(0.58) |
(2.44) |
(0.35) |
(1.87) |
(2.79) |
(2.05) |
(3.28) |
||
|
4-factor α |
-0.36 |
-0.29 |
-0.06 |
0.50 |
0.15 |
0.75 |
0.02 |
0.45 |
0.66 |
0.70 |
1.05 |
|
|
(-1.54) |
(-1.41) |
(-0.25) |
(1.72) |
(0.58) |
(2.55) |
(0.07) |
(1.95) |
(2.86) |
(2.91) |
(3.75) |
||
|
Sharpe Ratio |
0.21 |
0.38 |
0.48 |
0.56 |
0.56 |
0.57 |
0.56 |
0.58 |
0.53 |
0.55 |
0.59 |
|
|
Information Ratio |
(-0.38) |
(-0.37) |
(-0.07) |
(0.51) |
(0.17) |
(0.90) |
(0.02) |
(0.54) |
(0.83) |
(0.86) |
(1.04) |
|
|
Brazil |
Excess Return |
0.10 |
1.37 |
0.78 |
1.02 |
1.49 |
1.64 |
1.46 |
1.04 |
1.47 |
1.36 |
1.26 |
|
(0.17) |
(1.91) |
(1.21) |
(1.71) |
(2.36) |
(2.56) |
(2.46) |
(1.78) |
(2.61) |
(2.69) |
(3.87) |
||
|
CAPM α |
0.75 |
2.80 |
0.70 |
1.68 |
1.15 |
2.28 |
1.13 |
0.72 |
1.44 |
1.24 |
0.49 |
|
|
(1.14) |
(5.10) |
(1.08) |
(3.96) |
(2.28) |
(3.39) |
(1.75) |
(1.33) |
(2.73) |
(4.07) |
(0.64) |
||
|
3-factor α |
1.22 |
2.02 |
1.67 |
1.54 |
1.54 |
1.21 |
2.20 |
0.71 |
1.49 |
1.23 |
0.01 |
|
|
(2.24) |
(3.08) |
(1.14) |
(3.30) |
(2.71) |
(1.49) |
(3.64) |
(1.38) |
(3.20) |
(2.96) |
(0.02) |
||
|
4-factor α |
1.69 |
1.65 |
1.12 |
1.27 |
1.27 |
0.37 |
1.88 |
0.55 |
1.25 |
0.92 |
-0.76 |
|
|
(3.21) |
(2.11) |
(0.79) |
(2.31) |
(2.29) |
(0.40) |
(3.39) |
(1.05) |
(3.12) |
(2.25) |
(-1.13) |
||
|
Sharpe Ratio |
0.05 |
0.66 |
0.43 |
0.58 |
0.92 |
0.94 |
0.88 |
0.63 |
1.00 |
1.02 |
0.84 |
|
|
Information Ratio |
(0.71) |
(0.63) |
(0.22) |
(0.68) |
(0.65) |
(0.12) |
(1.02) |
(0.30) |
(0.84) |
(0.58) |
(-0.24) |
|
|
Netherlands |
Excess Return |
-0.70 |
0.21 |
0.27 |
0.45 |
0.27 |
0.90 |
0.64 |
1.13 |
0.31 |
1.08 |
1.78 |
|
(-1.23) |
(0.38) |
(0.51) |
(0.97) |
(0.62) |
(2.32) |
(1.49) |
(2.51) |
(0.72) |
(2.40) |
(4.46) |
||
|
CAPM α |
-0.84 |
0.05 |
0.43 |
0.83 |
0.09 |
1.12 |
0.59 |
1.58 |
1.12 |
0.37 |
1.21 |
|
|
(-1.50) |
(0.09) |
(0.76) |
(1.30) |
(0.16) |
(2.52) |
(1.05) |
(2.71) |
(2.07) |
(0.61) |
(1.42) |
||
|
3-factor α |
-0.40 |
1.10 |
1.00 |
0.99 |
0.41 |
0.93 |
0.44 |
0.78 |
1.37 |
0.08 |
0.48 |
|
|
(-0.46) |
(1.49) |
(1.75) |
(1.37) |
(0.76) |
(1.77) |
(0.64) |
(1.33) |
(2.64) |
(0.15) |
(0.49) |
||
|
4-factor α |
-1.17 |
0.87 |
0.09 |
1.81 |
0.12 |
1.39 |
0.20 |
-0.26 |
0.83 |
0.34 |
0.00 |
|
|
(-1.13) |
(0.94) |
(0.15) |
(2.23) |
(0.22) |
(1.72) |
(0.26) |
(-0.34) |
(1.36) |
(0.51) |
(1.30) |
||
|
Sharpe Ratio |
-0.37 |
0.11 |
0.15 |
0.27 |
0.17 |
0.63 |
0.44 |
0.74 |
0.24 |
0.81 |
1.39 |
|
|
Information Ratio |
(-0.27) |
(0.22) |
(0.03) |
(0.68) |
(0.06) |
(0.49) |
(0.07) |
(-0.08) |
(0.39) |
(0.14) |
(0.37) |
|
|
South Africa |
Excess Return |
0.60 |
0.64 |
1.06 |
0.95 |
0.76 |
1.01 |
0.88 |
1.19 |
1.12 |
1.38 |
0.78 |
|
(1.20) |
(1.76) |
(3.03) |
(2.62) |
(1.94) |
(2.79) |
(2.52) |
(3.59) |
(3.27) |
(4.32) |
(2.04) |
||
|
CAPM α |
0.32 |
1.01 |
1.43 |
0.94 |
0.95 |
0.53 |
0.99 |
1.97 |
0.97 |
1.52 |
1.20 |
|
|
(0.53) |
(2.83) |
(2.87) |
(2.82) |
(1.95) |
(1.19) |
(2.45) |
(4.80) |
(3.09) |
(2.99) |
(2.09) |
||
|
3-factor α |
-0.02 |
1.25 |
0.98 |
0.88 |
1.22 |
-0.01 |
0.76 |
1.90 |
1.07 |
1.18 |
1.20 |
|
|
(-0.05) |
(3.00) |
(2.20) |
(2.57) |
(2.97) |
(-0.01) |
(2.07) |
(4.33) |
(3.50) |
(3.09) |
(2.59) |
||
|
4-factor α |
0.00 |
1.34 |
1.00 |
0.77 |
1.20 |
0.07 |
0.86 |
1.87 |
0.89 |
1.17 |
1.17 |
|
|
(0.01) |
(3.40) |
(2.15) |
(2.19) |
(2.93) |
(0.15) |
(2.43) |
(4.28) |
(2.61) |
(3.22) |
(2.68) |
||
|
Sharpe Ratio |
0.45 |
0.54 |
1.11 |
0.87 |
0.66 |
1.03 |
0.88 |
1.22 |
1.17 |
1.54 |
0.63 |
|
|
Information Ratio |
(0.00) |
(0.70) |
(0.59) |
(0.47) |
(0.90) |
(0.05) |
(0.69) |
(1.38) |
(0.77) |
(1.04) |
(0.62) |
|
|
Singapore |
Excess Return |
0.40 |
0.58 |
0.71 |
0.68 |
0.61 |
0.66 |
0.86 |
0.87 |
0.74 |
0.96 |
0.56 |
|
(0.57) |
(0.78) |
(1.13) |
(1.09) |
(1.00) |
(0.96) |
(1.44) |
(1.46) |
(1.33) |
(1.67) |
(1.52) |
||
|
CAPM α |
0.46 |
0.60 |
1.16 |
0.86 |
0.15 |
0.84 |
0.77 |
1.17 |
0.74 |
1.41 |
0.95 |
|
|
(1.19) |
(1.62) |
(2.42) |
(2.11) |
(0.43) |
(2.17) |
(2.65) |
(3.99) |
(3.16) |
(4.24) |
(1.93) |
||
|
3-factor α |
0.23 |
1.52 |
0.91 |
0.74 |
0.14 |
0.87 |
0.72 |
1.27 |
0.65 |
1.29 |
1.06 |
|
|
(0.65) |
(1.60) |
(2.09) |
(2.08) |
(0.38) |
(2.56) |
(2.52) |
(3.48) |
(2.91) |
(4.21) |
(2.31) |
||
|
4-factor α |
0.44 |
0.66 |
0.68 |
0.71 |
-0.27 |
0.59 |
0.74 |
0.90 |
0.60 |
1.08 |
0.65 |
|
|
(0.99) |
(1.76) |
(1.81) |
(2.08) |
(-0.71) |
(1.93) |
(2.54) |
(2.69) |
(2.45) |
(3.66) |
(1.38) |
||
|
Sharpe Ratio |
0.18 |
0.27 |
0.38 |
0.37 |
0.35 |
0.36 |
0.51 |
0.53 |
0.47 |
0.59 |
0.42 |
|
|
Information Ratio |
(0.24) |
(0.40) |
(0.50) |
(0.56) |
(-0.18) |
(0.50) |
(0.62) |
(0.83) |
(0.56) |
(0.97) |
(0.32) |
|
|
Mexico |
Excess Return |
0.18 |
0.17 |
0.37 |
0.82 |
0.50 |
1.07 |
1.20 |
0.87 |
1.00 |
1.09 |
0.91 |
|
(0.43) |
(0.43) |
(0.89) |
(2.66) |
(1.38) |
(3.58) |
(3.95) |
(2.91) |
(3.00) |
(0.00) |
(2.83) |
||
|
CAPM α |
0.68 |
0.30 |
0.55 |
0.91 |
1.01 |
1.24 |
0.93 |
0.76 |
0.39 |
0.95 |
0.27 |
|
|
(1.34) |
(0.91) |
(1.67) |
(3.56) |
(2.63) |
(0.00) |
(3.34) |
(2.44) |
(0.94) |
(3.28) |
(0.47) |
||
|
3-factor α |
1.38 |
0.78 |
0.83 |
0.80 |
1.45 |
1.12 |
0.78 |
0.88 |
0.61 |
0.74 |
-0.64 |
|
|
(2.92) |
(2.65) |
(0.00) |
(3.11) |
(3.92) |
(0.00) |
(2.18) |
(2.79) |
(2.75) |
(2.26) |
(-1.11) |
||
|
4-factor α |
1.29 |
-0.64 |
0.31 |
0.81 |
1.44 |
0.95 |
0.75 |
0.77 |
0.44 |
0.15 |
-1.14 |
|
|
(2.27) |
(-1.17) |
(0.85) |
(2.53) |
(2.51) |
(2.82) |
(1.77) |
(2.85) |
(1.61) |
(0.37) |
(-1.58) |
||
|
Sharpe Ratio |
0.14 |
0.14 |
0.33 |
0.87 |
0.49 |
1.19 |
1.41 |
0.94 |
0.94 |
0.99 |
0.78 |
|
|
Information Ratio |
(0.66) |
(-0.35) |
(0.21) |
(0.57) |
(0.97) |
(0.78) |
(0.66) |
(0.61) |
(0.39) |
(0.11) |
(-0.44) |
|
|
Norway |
Excess Return |
-0.35 |
-0.51 |
0.34 |
0.25 |
0.22 |
0.33 |
0.70 |
0.64 |
0.67 |
0.92 |
1.27 |
|
(-0.54) |
(-0.83) |
(0.58) |
(0.46) |
(0.41) |
(0.72) |
(1.34) |
(1.46) |
(1.48) |
(1.89) |
(3.38) |
||
|
CAPM α |
-1.45 |
-3.84 |
0.87 |
0.64 |
0.57 |
0.47 |
0.84 |
1.17 |
0.52 |
1.31 |
2.76 |
|
|
(-1.24) |
(-1.35) |
(1.71) |
(1.22) |
(1.42) |
(1.49) |
(2.21) |
(2.60) |
(0.72) |
(3.54) |
(2.28) |
||
|
3-factor α |
-0.36 |
-0.39 |
0.90 |
0.36 |
1.04 |
0.77 |
1.02 |
0.85 |
0.96 |
1.32 |
1.68 |
|
|
(-0.59) |
(-0.84) |
(1.96) |
(0.77) |
(2.77) |
(2.28) |
(2.86) |
(1.94) |
(2.22) |
(3.20) |
(2.38) |
||
|
4-factor α |
-0.64 |
-0.41 |
0.86 |
0.19 |
0.97 |
0.39 |
1.00 |
0.85 |
0.82 |
1.25 |
1.89 |
|
|
(-1.11) |
(-0.81) |
(1.71) |
(0.42) |
(2.64) |
(1.04) |
(2.45) |
(2.00) |
(1.82) |
(3.42) |
(2.87) |
||
|
Sharpe Ratio |
-0.18 |
-0.28 |
0.22 |
0.16 |
0.16 |
0.26 |
0.48 |
0.49 |
0.50 |
0.72 |
0.93 |
|
|
Information Ratio |
(-0.26) |
(-0.21) |
(0.50) |
(0.12) |
(0.75) |
(0.31) |
(0.64) |
(0.54) |
(0.51) |
(0.95) |
(0.73) |
|
|
Indonesia |
Excess Return |
1.03 |
1.68 |
1.88 |
1.61 |
1.83 |
1.64 |
1.38 |
1.60 |
1.94 |
1.82 |
0.79 |
|
(2.15) |
(3.36) |
(3.52) |
(3.86) |
(3.70) |
(3.39) |
(3.08) |
(3.07) |
(3.57) |
(3.14) |
(2.08) |
||
|
CAPM α |
1.38 |
1.32 |
1.14 |
1.46 |
2.06 |
1.61 |
1.53 |
1.47 |
2.81 |
1.80 |
0.42 |
|
|
(2.85) |
(2.82) |
(3.50) |
(4.84) |
(4.25) |
(4.39) |
(4.00) |
(3.54) |
(3.54) |
(4.13) |
(0.89) |
||
|
3-factor α |
1.30 |
0.95 |
0.85 |
0.99 |
1.75 |
1.55 |
1.45 |
1.19 |
2.17 |
1.95 |
0.64 |
|
|
(3.35) |
(2.46) |
(2.46) |
(3.21) |
(4.39) |
(4.01) |
(4.17) |
(3.03) |
(5.13) |
(5.49) |
(1.42) |
||
|
4-factor α |
1.17 |
0.97 |
0.69 |
0.98 |
1.61 |
1.51 |
1.37 |
1.19 |
2.19 |
1.80 |
0.63 |
|
|
(3.16) |
(2.46) |
(1.87) |
(3.09) |
(4.42) |
(3.82) |
(3.53) |
(3.47) |
(5.45) |
(5.73) |
(1.34) |
||
|
Sharpe Ratio |
0.73 |
1.18 |
1.27 |
1.20 |
1.22 |
1.16 |
0.94 |
1.11 |
1.19 |
1.19 |
0.67 |
|
|
Information Ratio |
(0.95) |
(0.65) |
(0.53) |
(0.83) |
(1.23) |
(1.20) |
(0.99) |
(0.99) |
(1.79) |
(1.41) |
(0.35) |
|
|
Denmark |
Excess Return |
-0.64 |
0.28 |
-0.10 |
-0.13 |
0.19 |
0.27 |
0.70 |
0.58 |
0.69 |
0.96 |
1.61 |
|
(-1.06) |
(0.44) |
(-0.18) |
(-0.23) |
(0.35) |
(0.60) |
(1.46) |
(1.13) |
(1.55) |
(2.08) |
(3.97) |
||
|
CAPM α |
-0.80 |
0.38 |
-0.22 |
-0.14 |
0.25 |
0.87 |
0.65 |
0.26 |
0.79 |
0.85 |
1.65 |
|
|
(-1.53) |
(0.66) |
(-0.42) |
(-0.29) |
(0.61) |
(1.63) |
(1.27) |
(0.56) |
(1.74) |
(1.87) |
(2.90) |
||
|
3-factor α |
-0.37 |
1.38 |
0.48 |
-0.01 |
0.39 |
0.46 |
1.23 |
0.49 |
1.07 |
0.51 |
0.89 |
|
|
(-0.44) |
(2.20) |
(1.22) |
(-0.02) |
(0.79) |
(0.78) |
(2.72) |
(1.09) |
(2.88) |
(1.06) |
(0.92) |
||
|
4-factor α |
-0.36 |
1.19 |
0.77 |
-0.28 |
0.35 |
0.65 |
0.86 |
0.31 |
0.54 |
0.43 |
0.79 |
|
|
(-0.40) |
(1.93) |
(1.49) |
(-0.48) |
(0.61) |
(1.18) |
(1.63) |
(0.65) |
(1.27) |
(0.86) |
(0.76) |
||
|
Sharpe Ratio |
-0.35 |
0.17 |
-0.07 |
-0.10 |
0.14 |
0.19 |
0.54 |
0.44 |
0.57 |
0.74 |
1.24 |
|
|
Information Ratio |
(-0.12) |
(0.51) |
(0.36) |
(-0.13) |
(0.21) |
(0.33) |
(0.49) |
(0.18) |
(0.33) |
(0.25) |
(0.25) |
|
|
Thailand |
Excess Return |
0.57 |
0.76 |
0.78 |
0.82 |
0.93 |
1.04 |
1.07 |
0.90 |
0.93 |
1.02 |
0.45 |
|
(1.28) |
(1.55) |
(1.69) |
(1.85) |
(1.88) |
(2.25) |
(2.18) |
(1.98) |
(2.21) |
(2.38) |
(2.13) |
||
|
CAPM α |
0.30 |
0.65 |
0.62 |
0.29 |
1.02 |
0.80 |
0.99 |
0.97 |
0.92 |
1.03 |
0.73 |
|
|
(1.03) |
(1.80) |
(2.22) |
(1.10) |
(4.01) |
(3.12) |
(3.87) |
(3.92) |
(3.89) |
(4.39) |
(2.46) |
||
|
3-factor α |
0.22 |
0.60 |
0.83 |
0.75 |
1.02 |
0.83 |
0.86 |
1.36 |
1.26 |
1.23 |
1.00 |
|
|
(0.84) |
(2.18) |
(2.97) |
(2.73) |
(4.20) |
(3.48) |
(3.24) |
(5.48) |
(4.88) |
(5.47) |
(2.96) |
||
|
4-factor α |
0.29 |
0.58 |
0.72 |
0.61 |
0.87 |
0.74 |
0.63 |
1.24 |
1.20 |
1.15 |
0.85 |
|
|
(1.06) |
(2.05) |
(2.45) |
(2.17) |
(3.61) |
(3.37) |
(2.51) |
(5.12) |
(5.05) |
(5.38) |
(2.67) |
||
|
Sharpe Ratio |
0.41 |
0.52 |
0.56 |
0.63 |
0.72 |
0.84 |
0.83 |
0.73 |
0.80 |
0.89 |
0.59 |
|
|
Information Ratio |
(0.32) |
(0.61) |
(0.75) |
(0.66) |
(0.94) |
(0.91) |
(0.78) |
(1.48) |
(1.67) |
(1.59) |
(0.82) |
|
|
Finland |
Excess Return |
-0.55 |
0.02 |
-0.01 |
0.40 |
0.77 |
0.81 |
0.63 |
0.95 |
0.65 |
0.92 |
1.47 |
|
(-1.04) |
(0.05) |
(-0.02) |
(0.84) |
(1.65) |
(1.87) |
(1.48) |
(2.22) |
(1.47) |
(2.35) |
(3.95) |
||
|
CAPM α |
-0.72 |
-1.29 |
-1.16 |
-0.75 |
0.54 |
0.79 |
0.69 |
1.07 |
-0.97 |
0.65 |
1.38 |
|
|
(-0.85) |
(-1.47) |
(-1.46) |
(-1.38) |
(1.11) |
(1.34) |
(1.77) |
(2.20) |
(-0.95) |
(1.29) |
(1.39) |
||
|
3-factor α |
0.91 |
-0.07 |
0.45 |
0.39 |
0.87 |
0.76 |
1.42 |
1.08 |
0.07 |
1.10 |
0.19 |
|
|
(0.93) |
(-0.10) |
(0.53) |
(0.69) |
(1.75) |
(1.09) |
(3.10) |
(2.01) |
(0.13) |
(2.09) |
(0.17) |
||
|
4-factor α |
1.01 |
0.17 |
1.05 |
0.21 |
1.37 |
0.46 |
1.24 |
1.32 |
0.23 |
0.89 |
-0.12 |
|
|
(1.05) |
(0.24) |
(1.33) |
(0.32) |
(1.73) |
(0.54) |
(2.24) |
(2.11) |
(0.34) |
(1.69) |
(-0.12) |
||
|
Sharpe Ratio |
-0.31 |
0.01 |
-0.01 |
0.26 |
0.54 |
0.60 |
0.44 |
0.64 |
0.44 |
0.69 |
1.22 |
|
|
Information Ratio |
(0.27) |
(0.06) |
(0.40) |
(0.09) |
(0.66) |
(0.19) |
(0.62) |
(0.64) |
(0.11) |
(0.42) |
(-0.03) |
|
|
Turkey |
Excess Return |
0.86 |
1.08 |
1.36 |
1.35 |
1.46 |
1.38 |
1.63 |
1.51 |
1.58 |
1.86 |
1.00 |
|
(1.40) |
(1.68) |
(2.25) |
(2.30) |
(2.60) |
(2.11) |
(2.58) |
(2.46) |
(2.52) |
(3.23) |
(3.04) |
||
|
CAPM α |
-0.48 |
0.42 |
1.63 |
-1.07 |
3.85 |
0.50 |
0.96 |
1.63 |
0.53 |
2.83 |
3.31 |
|
|
(-0.70) |
(0.72) |
(2.33) |
(-0.45) |
(2.09) |
(0.80) |
(1.63) |
(2.32) |
(0.77) |
(2.10) |
(2.31) |
||
|
3-factor α |
-0.14 |
0.42 |
2.37 |
1.45 |
2.16 |
0.73 |
2.04 |
1.90 |
0.16 |
2.73 |
2.88 |
|
|
(-0.22) |
(0.64) |
(3.52) |
(2.22) |
(3.26) |
(1.29) |
(3.21) |
(2.67) |
(0.25) |
(2.79) |
(2.61) |
||
|
4-factor α |
0.07 |
0.46 |
2.74 |
0.90 |
2.42 |
0.24 |
1.55 |
1.78 |
0.16 |
1.94 |
1.87 |
|
|
(0.11) |
(0.62) |
(3.39) |
(1.30) |
(3.50) |
(0.38) |
(2.49) |
(2.62) |
(0.24) |
(3.13) |
(1.96) |
||
|
Sharpe Ratio |
0.46 |
0.55 |
0.72 |
0.77 |
0.75 |
0.70 |
0.86 |
0.80 |
0.86 |
1.08 |
0.86 |
|
|
Information Ratio |
(0.03) |
(0.17) |
(1.16) |
(0.43) |
(1.13) |
(0.09) |
(0.66) |
(0.68) |
(0.07) |
(0.96) |
(0.58) |
|
|
Chile |
Excess Return |
0.49 |
0.70 |
0.71 |
0.69 |
0.75 |
0.81 |
0.77 |
0.75 |
0.91 |
1.22 |
0.73 |
|
(1.46) |
(2.00) |
(2.66) |
(2.35) |
(2.90) |
(3.39) |
(3.51) |
(3.28) |
(3.89) |
(4.54) |
(3.41) |
||
|
CAPM α |
0.27 |
0.31 |
0.49 |
0.52 |
0.64 |
0.69 |
0.73 |
0.70 |
0.92 |
1.07 |
0.80 |
|
|
(1.30) |
(1.38) |
(2.03) |
(2.81) |
(3.24) |
(3.30) |
(3.56) |
(4.22) |
(5.68) |
(3.97) |
(2.93) |
||
|
3-factor α |
0.26 |
0.25 |
0.46 |
0.20 |
0.83 |
0.52 |
0.75 |
0.89 |
1.08 |
1.16 |
0.90 |
|
|
(1.35) |
(1.08) |
(1.85) |
(1.21) |
(3.42) |
(2.02) |
(4.52) |
(5.55) |
(6.73) |
(4.89) |
(3.21) |
||
|
4-factor α |
0.38 |
0.22 |
0.47 |
0.06 |
0.82 |
0.45 |
0.64 |
0.94 |
1.01 |
1.15 |
0.78 |
|
|
(1.73) |
(0.95) |
(1.62) |
(0.34) |
(3.42) |
(1.69) |
(3.76) |
(5.67) |
(5.91) |
(4.57) |
(2.48) |
||
|
Sharpe Ratio |
0.50 |
0.74 |
0.86 |
0.88 |
0.97 |
1.05 |
1.17 |
1.06 |
1.31 |
1.54 |
0.83 |
|
|
Information Ratio |
(0.37) |
(0.24) |
(0.48) |
(0.09) |
(1.06) |
(0.61) |
(0.97) |
(1.43) |
(1.31) |
(1.38) |
(0.61) |
|
|
Poland |
Excess Return |
0.47 |
0.57 |
0.74 |
0.61 |
0.56 |
0.88 |
0.49 |
0.63 |
0.82 |
0.76 |
0.30 |
|
(0.53) |
(0.78) |
(1.09) |
(0.82) |
(0.92) |
(1.35) |
(0.71) |
(0.93) |
(1.40) |
(1.33) |
(0.60) |
||
|
CAPM α |
-1.09 |
0.42 |
0.56 |
0.55 |
0.36 |
0.60 |
0.73 |
-0.10 |
0.87 |
0.93 |
2.01 |
|
|
(-1.50) |
(0.66) |
(1.27) |
(0.90) |
(0.78) |
(1.29) |
(1.35) |
(-0.24) |
(2.09) |
(2.11) |
(2.77) |
||
|
3-factor α |
-0.94 |
0.85 |
0.61 |
0.52 |
0.20 |
0.41 |
1.04 |
0.50 |
0.86 |
0.68 |
1.61 |
|
|
(-1.46) |
(1.58) |
(1.33) |
(0.81) |
(0.46) |
(0.84) |
(1.76) |
(1.08) |
(1.84) |
(2.41) |
(2.40) |
||
|
4-factor α |
-1.09 |
0.86 |
0.67 |
-2.19 |
0.06 |
0.01 |
0.70 |
0.47 |
0.69 |
0.41 |
1.50 |
|
|
(-1.71) |
(1.72) |
(1.38) |
(-0.79) |
(0.11) |
(0.01) |
(1.14) |
(0.88) |
(1.61) |
(1.30) |
(2.22) |
||
|
Sharpe Ratio |
0.19 |
0.29 |
0.40 |
0.31 |
0.30 |
0.51 |
0.27 |
0.38 |
0.50 |
0.52 |
0.20 |
|
|
Information Ratio |
(-0.39) |
(0.51) |
(0.37) |
(-0.20) |
(0.03) |
(0.00) |
(0.37) |
(0.25) |
(0.40) |
(0.34) |
(0.58) |
|
|
Colombia |
Excess Return |
1.45 |
1.24 |
1.02 |
1.02 |
0.71 |
0.81 |
1.09 |
1.42 |
1.48 |
1.48 |
0.03 |
|
(3.08) |
(2.19) |
(2.17) |
(2.57) |
(2.00) |
(1.70) |
(1.84) |
(3.85) |
(3.25) |
(2.96) |
(0.07) |
||
|
CAPM α |
1.35 |
0.01 |
0.30 |
1.42 |
0.08 |
0.30 |
1.77 |
3.42 |
2.00 |
1.78 |
0.44 |
|
|
(1.63) |
(0.02) |
(0.61) |
(1.57) |
(0.13) |
(0.68) |
(2.36) |
(1.46) |
(3.02) |
(3.02) |
(0.43) |
||
|
3-factor α |
-56.24 |
1.09 |
0.87 |
0.78 |
-7.33 |
-0.67 |
0.61 |
-1.90 |
4.11 |
-1.13 |
55.11 |
|
|
(-0.98) |
(1.01) |
(1.37) |
(1.54) |
(-0.75) |
(-0.57) |
(0.57) |
(-0.63) |
(1.66) |
(-0.60) |
(0.96) |
||
|
4-factor α |
-3.18 |
-0.75 |
0.60 |
4.02 |
5.93 |
9.80 |
2.37 |
1.07 |
4.07 |
1.15 |
4.32 |
|
|
(-0.47) |
(-0.37) |
(0.36) |
(1.29) |
(0.94) |
(1.45) |
(1.54) |
(0.62) |
(1.62) |
(1.70) |
(0.62) |
||
|
Sharpe Ratio |
0.91 |
0.77 |
0.68 |
0.68 |
0.55 |
0.52 |
0.74 |
1.16 |
1.01 |
0.94 |
0.02 |
|
|
Information Ratio |
(-0.08) |
(-0.09) |
(0.07) |
(0.42) |
(0.39) |
(0.74) |
(0.41) |
(0.21) |
(0.58) |
(0.48) |
(0.16) |
|
|
Austria |
Excess Return |
0.06 |
0.41 |
-0.06 |
0.56 |
0.52 |
0.81 |
0.68 |
0.63 |
1.05 |
0.35 |
0.29 |
|
(0.09) |
(0.86) |
(-0.12) |
(1.09) |
(0.96) |
(1.56) |
(1.42) |
(1.45) |
(2.79) |
(0.79) |
(0.59) |
||
|
CAPM α |
-0.24 |
-0.36 |
-0.14 |
1.28 |
1.08 |
2.82 |
0.52 |
0.58 |
0.78 |
0.76 |
1.00 |
|
|
(-0.31) |
(-0.75) |
(-0.28) |
(1.31) |
(1.70) |
(1.17) |
(0.99) |
(1.05) |
(1.99) |
(1.29) |
(1.25) |
||
|
3-factor α |
1.83 |
-0.42 |
0.17 |
0.36 |
1.64 |
1.38 |
-0.22 |
0.06 |
-0.06 |
0.28 |
-1.55 |
|
|
(1.54) |
(-0.97) |
(0.22) |
(0.47) |
(1.56) |
(2.24) |
(-0.27) |
(0.09) |
(-0.11) |
(0.54) |
(-1.29) |
||
|
4-factor α |
1.05 |
-4.74 |
3.20 |
-1.39 |
0.30 |
2.11 |
1.11 |
-1.66 |
0.18 |
-1.05 |
-2.10 |
|
|
(0.94) |
(-1.10) |
(1.24) |
(-1.47) |
(1.05) |
(2.04) |
(1.25) |
(-1.76) |
(0.17) |
(-1.24) |
(-1.55) |
||
|
Sharpe Ratio |
0.03 |
0.26 |
-0.04 |
0.32 |
0.32 |
0.49 |
0.43 |
0.44 |
0.74 |
0.25 |
0.15 |
|
|
Information Ratio |
(0.28) |
(-0.23) |
(0.50) |
(-0.35) |
(0.29) |
(0.61) |
(0.34) |
(-0.33) |
(0.05) |
(-0.26) |
(-0.35) |
|
|
Philippines |
Excess Return |
1.29 |
1.80 |
1.58 |
1.46 |
0.84 |
1.70 |
1.76 |
1.76 |
1.39 |
1.74 |
0.45 |
|
(2.04) |
(3.19) |
(3.04) |
(2.91) |
(1.39) |
(3.07) |
(3.06) |
(3.18) |
(3.27) |
(3.32) |
(1.02) |
||
|
CAPM α |
1.04 |
1.72 |
1.94 |
1.21 |
0.31 |
1.83 |
4.18 |
1.22 |
2.10 |
1.67 |
0.63 |
|
|
(2.07) |
(2.97) |
(3.50) |
(2.02) |
(0.57) |
(3.15) |
(2.01) |
(2.11) |
(4.44) |
(3.98) |
(0.96) |
||
|
3-factor α |
0.37 |
1.45 |
1.74 |
1.27 |
-0.33 |
1.83 |
1.90 |
1.50 |
1.94 |
1.56 |
1.19 |
|
|
(1.01) |
(2.81) |
(3.55) |
(2.24) |
(-0.41) |
(2.85) |
(3.31) |
(3.09) |
(3.91) |
(3.72) |
(2.20) |
||
|
4-factor α |
0.33 |
1.55 |
1.64 |
1.79 |
0.00 |
1.66 |
1.81 |
1.40 |
2.00 |
1.56 |
1.22 |
|
|
(0.99) |
(2.77) |
(3.11) |
(2.25) |
(0.01) |
(2.53) |
(2.88) |
(2.73) |
(4.04) |
(3.32) |
(2.15) |
||
|
Sharpe Ratio |
0.70 |
1.06 |
1.06 |
0.95 |
0.49 |
1.09 |
1.11 |
1.15 |
1.07 |
1.28 |
0.27 |
|
|
Information Ratio |
(0.21) |
(0.76) |
(0.80) |
(0.66) |
(0.00) |
(0.71) |
(0.91) |
(0.72) |
(1.23) |
(0.87) |
(0.54) |
|
|
Argentina |
Excess Return |
1.92 |
1.82 |
2.49 |
1.79 |
2.67 |
2.69 |
2.68 |
2.51 |
1.87 |
2.33 |
0.40 |
|
(2.27) |
(2.49) |
(3.20) |
(2.34) |
(3.40) |
(3.76) |
(3.52) |
(4.08) |
(2.95) |
(4.03) |
(0.68) |
||
|
CAPM α |
-0.24 |
-0.78 |
0.24 |
1.95 |
1.94 |
1.14 |
3.73 |
1.92 |
1.95 |
1.79 |
2.03 |
|
|
(-0.19) |
(-0.71) |
(0.24) |
(2.39) |
(1.96) |
(1.56) |
(3.93) |
(1.86) |
(2.42) |
(2.33) |
(1.29) |
||
|
3-factor α |
0.06 |
-1.71 |
-0.62 |
1.45 |
2.52 |
0.76 |
5.44 |
2.08 |
4.94 |
1.57 |
1.52 |
|
|
(0.03) |
(-1.38) |
(-0.46) |
(1.19) |
(1.42) |
(0.89) |
(3.98) |
(1.40) |
(1.96) |
(1.65) |
(0.63) |
||
|
4-factor α |
-0.33 |
-1.08 |
0.15 |
3.67 |
3.43 |
1.09 |
5.31 |
3.55 |
0.94 |
2.71 |
3.05 |
|
|
(-0.17) |
(-0.76) |
(0.10) |
(2.35) |
(1.70) |
(0.99) |
(3.72) |
(1.98) |
(0.60) |
(2.14) |
(1.14) |
||
|
Sharpe Ratio |
0.71 |
0.80 |
1.13 |
0.81 |
1.27 |
1.42 |
1.32 |
1.27 |
0.98 |
1.33 |
0.17 |
|
|
Information Ratio |
(-0.05) |
(-0.20) |
(0.03) |
(0.86) |
(0.57) |
(0.29) |
(1.44) |
(0.80) |
(0.16) |
(0.73) |
(0.42) |
|
|
Ireland |
Excess Return |
0.46 |
1.49 |
0.97 |
-0.19 |
0.03 |
-0.34 |
0.34 |
-0.55 |
-0.06 |
0.75 |
0.28 |
|
(0.54) |
(1.71) |
(1.32) |
(-0.19) |
(0.03) |
(-0.34) |
(0.40) |
(-0.78) |
(-0.07) |
(1.06) |
(0.34) |
||
|
CAPM α |
2.17 |
14.90 |
3.14 |
2.91 |
0.37 |
0.00 |
-1.31 |
-3.55 |
8.18 |
-0.33 |
-2.50 |
|
|
(0.92) |
(1.08) |
(1.25) |
(1.45) |
(0.30) |
(1.12) |
(-0.31) |
(-2.05) |
(1.07) |
(-0.24) |
(-0.88) |
||
|
3-factor α |
2.22 |
4.98 |
1.95 |
20.17 |
4.16 |
-1.36 |
17.18 |
-0.76 |
3.94 |
10.02 |
7.80 |
|
|
(0.58) |
(2.28) |
(0.50) |
(1.13) |
(0.63) |
(-0.26) |
(1.33) |
(-0.35) |
(0.69) |
(1.49) |
(1.04) |
||
|
4-factor α |
-0.33 |
1.24 |
-0.47 |
3.40 |
1.28 |
1.56 |
-10.30 |
-0.36 |
-1.05 |
-5.25 |
-4.91 |
|
|
(-0.17) |
(1.02) |
(-0.27) |
(1.90) |
(0.70) |
(0.00) |
(-0.95) |
(-0.30) |
(-0.59) |
(-0.98) |
(-0.88) |
||
|
Sharpe Ratio |
0.16 |
0.54 |
0.40 |
-0.06 |
0.01 |
-0.11 |
0.13 |
-0.22 |
-0.02 |
0.35 |
0.09 |
|
|
Information Ratio |
(-0.04) |
(0.25) |
(-0.06) |
(0.61) |
(0.15) |
(0.34) |
(-0.13) |
(-0.06) |
(-0.15) |
(-0.22) |
(-0.20) |
|
|
Greece |
Excess Return |
-0.33 |
0.54 |
0.02 |
0.38 |
0.22 |
0.12 |
0.44 |
0.65 |
0.42 |
0.60 |
0.92 |
|
(-0.45) |
(0.88) |
(0.03) |
(0.57) |
(0.33) |
(0.19) |
(0.64) |
(1.21) |
(0.81) |
(1.13) |
(2.32) |
||
|
CAPM α |
-0.23 |
0.49 |
-0.27 |
0.80 |
0.22 |
0.28 |
0.88 |
0.78 |
0.52 |
0.40 |
0.62 |
|
|
(-0.33) |
(0.80) |
(-0.50) |
(1.19) |
(0.36) |
(0.51) |
(1.73) |
(2.08) |
(1.23) |
(1.05) |
(0.96) |
||
|
3-factor α |
0.10 |
-0.92 |
0.22 |
0.44 |
-0.20 |
0.14 |
1.23 |
0.23 |
0.24 |
0.50 |
0.40 |
|
|
(0.13) |
(-1.19) |
(0.33) |
(0.62) |
(-0.41) |
(0.31) |
(2.54) |
(0.60) |
(0.58) |
(1.42) |
(0.45) |
||
|
4-factor α |
0.16 |
-0.70 |
0.66 |
0.66 |
-0.48 |
0.06 |
0.92 |
-0.01 |
0.07 |
0.34 |
0.18 |
|
|
(0.18) |
(-1.07) |
(0.89) |
(0.90) |
(-0.87) |
(0.12) |
(1.64) |
(-0.02) |
(0.18) |
(1.00) |
(0.19) |
||
|
Sharpe Ratio |
-0.14 |
0.26 |
0.01 |
0.17 |
0.10 |
0.06 |
0.22 |
0.41 |
0.26 |
0.38 |
0.55 |
|
|
Information Ratio |
(0.04) |
(-0.22) |
(0.25) |
(0.25) |
(-0.17) |
(0.03) |
(0.46) |
(-0.00) |
(0.04) |
(0.25) |
(0.05) |
|
|
Portugal |
Excess Return |
0.40 |
-0.09 |
0.35 |
1.07 |
0.11 |
0.78 |
1.53 |
0.76 |
0.61 |
1.40 |
1.00 |
|
(0.57) |
(-0.13) |
(0.62) |
(1.32) |
(0.19) |
(1.16) |
(2.46) |
(1.44) |
(1.30) |
(2.46) |
(1.43) |
||
|
CAPM α |
1.69 |
-0.42 |
0.36 |
1.59 |
0.16 |
1.45 |
1.02 |
0.86 |
0.39 |
2.60 |
0.91 |
|
|
(1.30) |
(-0.40) |
(0.35) |
(1.49) |
(0.17) |
(1.25) |
(1.05) |
(0.92) |
(0.38) |
(2.93) |
(0.66) |
||
|
3-factor α |
1.04 |
-9.34 |
-0.64 |
2.98 |
-0.49 |
12.82 |
3.40 |
22.08 |
6.13 |
0.58 |
-0.46 |
|
|
(0.68) |
(-1.27) |
(-0.34) |
(0.91) |
(-0.07) |
(1.48) |
(0.61) |
(0.91) |
(1.42) |
(0.30) |
(-0.17) |
||
|
4-factor α |
-1.05 |
0.08 |
46.63 |
-87.03 |
15.06 |
-16.29 |
0.13 |
-13.50 |
85.62 |
4.45 |
5.51 |
|
|
(-0.62) |
(0.00) |
(1.26) |
(-1.53) |
(1.18) |
(-1.27) |
(0.01) |
(-1.36) |
(0.86) |
(0.41) |
(0.47) |
||
|
Sharpe Ratio |
0.15 |
-0.04 |
0.17 |
0.42 |
0.06 |
0.37 |
0.74 |
0.42 |
0.34 |
0.63 |
0.33 |
|
|
Information Ratio |
(-0.15) |
(0.00) |
(3.24) |
(-0.04) |
(0.74) |
(-0.13) |
(0.00) |
(-0.24) |
(37.06) |
(0.10) |
(0.13) |
|
|
Peru |
Excess Return |
1.44 |
1.35 |
1.33 |
1.59 |
1.84 |
1.96 |
1.39 |
1.63 |
1.88 |
1.84 |
0.40 |
|
(2.85) |
(2.67) |
(2.54) |
(3.69) |
(4.28) |
(3.54) |
(2.49) |
(4.45) |
(4.48) |
(2.51) |
(0.58) |
||
|
CAPM α |
0.52 |
1.46 |
1.75 |
1.68 |
1.64 |
1.48 |
1.38 |
1.65 |
1.50 |
2.16 |
1.64 |
|
|
(1.49) |
(3.09) |
(2.49) |
(4.60) |
(3.55) |
(2.81) |
(3.23) |
(4.13) |
(4.20) |
(4.80) |
(3.19) |
||
|
3-factor α |
0.32 |
1.05 |
0.64 |
0.95 |
1.44 |
1.08 |
1.11 |
1.27 |
0.79 |
2.26 |
1.94 |
|
|
(0.64) |
(2.18) |
(1.69) |
(1.83) |
(2.34) |
(3.39) |
(2.69) |
(2.93) |
(1.43) |
(5.18) |
(3.34) |
||
|
4-factor α |
0.13 |
0.88 |
0.45 |
0.95 |
1.22 |
0.90 |
0.98 |
1.15 |
0.70 |
2.55 |
2.42 |
|
|
(0.24) |
(1.85) |
(1.16) |
(2.01) |
(2.28) |
(2.71) |
(2.56) |
(2.52) |
(1.21) |
(4.17) |
(3.54) |
||
|
Sharpe Ratio |
0.93 |
0.95 |
0.93 |
1.35 |
1.32 |
1.43 |
1.06 |
1.25 |
1.44 |
0.95 |
0.18 |
|
|
Information Ratio |
(0.06) |
(0.50) |
(0.29) |
(0.58) |
(0.62) |
(0.59) |
(0.61) |
(0.68) |
(0.41) |
(0.75) |
(0.68) |
|
|
Czech Republic |
Excess Return |
1.09 |
-0.88 |
0.93 |
0.54 |
0.91 |
0.59 |
0.57 |
1.38 |
1.06 |
1.01 |
-0.08 |
|
(1.84) |
(-1.72) |
(1.71) |
(1.05) |
(2.96) |
(1.63) |
(1.32) |
(3.37) |
(1.86) |
(1.80) |
(-0.10) |
||
|
CAPM α |
-0.19 |
-20.14 |
42.77 |
-81.77 |
57.81 |
-19.06 |
-80.13 |
12.42 |
13.36 |
2.36 |
2.55 |
|
|
(-0.19) |
(-0.58) |
(0.99) |
(-0.78) |
(1.34) |
(-0.57) |
(-0.86) |
(0.85) |
(0.15) |
(2.30) |
(1.84) |
||
|
3-factor α |
0.34 |
-0.15 |
0.94 |
0.97 |
-2.35 |
1.05 |
0.46 |
-0.01 |
0.77 |
2.57 |
2.23 |
|
|
(0.54) |
(-0.25) |
(2.14) |
(1.97) |
(-1.03) |
(1.85) |
(1.53) |
(-0.02) |
(1.04) |
(2.74) |
(1.81) |
||
|
4-factor α |
0.43 |
-0.15 |
0.67 |
1.07 |
0.30 |
0.61 |
0.39 |
0.37 |
0.78 |
1.97 |
1.53 |
|
|
(1.01) |
(-0.50) |
(1.75) |
(2.46) |
(0.75) |
(1.55) |
(1.40) |
(0.96) |
(1.28) |
(2.78) |
(1.72) |
||
|
Sharpe Ratio |
0.53 |
-0.38 |
0.42 |
0.21 |
0.62 |
0.42 |
0.26 |
0.92 |
0.55 |
0.60 |
-0.03 |
|
|
Information Ratio |
(0.30) |
(-0.11) |
(0.50) |
(0.66) |
(0.17) |
(0.42) |
(0.36) |
(0.24) |
(0.40) |
(0.77) |
(0.51) |
|
|
New Zealand |
Excess Return |
0.67 |
0.44 |
0.26 |
0.29 |
0.68 |
0.77 |
0.49 |
0.67 |
0.72 |
1.36 |
0.69 |
|
(1.20) |
(1.13) |
(0.71) |
(0.76) |
(1.88) |
(2.07) |
(1.33) |
(1.96) |
(1.96) |
(4.34) |
(1.26) |
||
|
CAPM α |
0.74 |
0.80 |
-0.29 |
1.14 |
0.84 |
0.72 |
0.44 |
0.65 |
0.19 |
1.48 |
0.74 |
|
|
(1.05) |
(1.29) |
(-0.42) |
(2.59) |
(1.60) |
(1.03) |
(0.55) |
(1.08) |
(0.34) |
(2.84) |
(0.87) |
||
|
3-factor α |
-0.28 |
0.40 |
-0.35 |
0.58 |
0.28 |
0.80 |
0.94 |
1.31 |
0.23 |
2.65 |
2.93 |
|
|
(-0.36) |
(0.65) |
(-0.70) |
(0.72) |
(0.63) |
(1.26) |
(1.59) |
(1.77) |
(0.42) |
(3.67) |
(2.68) |
||
|
4-factor α |
-0.14 |
-0.19 |
-0.21 |
0.91 |
0.30 |
0.80 |
0.57 |
0.62 |
0.14 |
1.74 |
1.88 |
|
|
(-0.17) |
(-0.30) |
(-0.31) |
(1.66) |
(0.62) |
(1.45) |
(1.04) |
(0.98) |
(0.26) |
(3.25) |
(1.83) |
||
|
Sharpe Ratio |
0.37 |
0.30 |
0.19 |
0.21 |
0.58 |
0.57 |
0.37 |
0.60 |
0.70 |
1.33 |
0.37 |
|
|
Information Ratio |
(-0.04) |
(-0.06) |
(-0.08) |
(0.40) |
(0.13) |
(0.40) |
(0.26) |
(0.25) |
(0.07) |
(0.90) |
(0.54) |
|
|
Pakistan |
Excess Return |
1.22 |
1.11 |
1.50 |
1.02 |
1.61 |
1.49 |
1.54 |
1.64 |
1.88 |
1.46 |
0.24 |
|
(1.63) |
(1.60) |
(2.40) |
(1.51) |
(2.32) |
(2.04) |
(2.13) |
(2.49) |
(2.93) |
(2.36) |
(0.46) |
||
|
CAPM α |
1.90 |
1.03 |
1.87 |
1.67 |
1.59 |
1.31 |
2.63 |
1.80 |
2.77 |
1.78 |
-0.12 |
|
|
(2.68) |
(1.79) |
(2.91) |
(2.79) |
(2.23) |
(2.25) |
(3.40) |
(3.41) |
(4.22) |
(3.58) |
(-0.19) |
||
|
3-factor α |
2.04 |
0.84 |
1.94 |
1.55 |
1.97 |
0.39 |
2.95 |
0.75 |
2.63 |
1.29 |
-0.75 |
|
|
(2.60) |
(1.40) |
(3.37) |
(2.88) |
(2.58) |
(0.71) |
(3.42) |
(1.32) |
(3.80) |
(2.71) |
(-0.89) |
||
|
4-factor α |
2.13 |
0.56 |
1.49 |
1.17 |
1.64 |
0.70 |
3.49 |
0.22 |
2.65 |
0.90 |
-1.23 |
|
|
(2.24) |
(0.99) |
(2.34) |
(2.10) |
(2.38) |
(0.99) |
(3.17) |
(0.33) |
(4.14) |
(1.84) |
(-1.17) |
||
|
Sharpe Ratio |
0.55 |
0.56 |
0.82 |
0.54 |
0.82 |
0.81 |
0.80 |
0.88 |
1.10 |
0.86 |
0.14 |
|
|
Information Ratio |
(0.71) |
(0.25) |
(0.73) |
(0.52) |
(0.68) |
(0.28) |
(1.42) |
(0.10) |
(1.26) |
(0.49) |
(-0.28) |